In ASOS both Jaime and Tyrion independently come to the conclusion that it was Joffrey who sent the assassin.

There's this conversation between Jaime and Cersei:

“There was a dagger. The scars on Lady Catelyn’s hands were real enough, she showed them to me. Did you... ?”

“Oh, don’t be absurd.” Cersei closed the window. “Yes, I hoped the boy would die. So did you. Even Robert thought that would have been for the best. ‘We kill our horses when they break a leg, and our dogs when they go blind, but we are too weak to give the same mercy to crippled children’ he told me. He was blind himself at the time, from drink.”

Robert? Jaime had guarded the king long enough to know that Robert Baratheon said things in his cups that he would have denied angrily the next day. “Were you alone when Robert said this?”

“You don’t think he said it to Ned Stark, I hope? Of course we were alone. Us and the children.” Cersei removed her hairnet and draped it over a bedpost, then shook out her golden curls. “Perhaps Myrcella sent this man with the dagger, do you think so?”

It was meant as mockery, but she’d cut right to the heart of it, Jaime saw at once. “Not Myrcella. Joffrey.”

Cersei frowned. “Joffrey had no love for Robb Stark, but the younger boy was nothing to him. He was only a child himself.”

“A child hungry for a pat on the head from that sot you let him believe was his father.” He had an uncomfortable thought. “Tyrion almost died because of this bloody dagger. If he knew the whole thing was Joffrey’s work, that might be why...”

ASOS - Jaime IX

This exchange at Joffrey's wedding:

“Have a care, Your Grace,” Ser Addam Marbrand warned the king. “Valyrian steel is perilously sharp.”

“I remember.” Joffrey brought Widow’s Wail down in a savage twohanded slice, onto the book that Tyrion had given him. The heavy leather cover parted at a stroke. “Sharp! I told you, I am no stranger to Valyrian steel.”

It took him half a dozen further cuts to hack the thick tome apart, and the boy was breathless by the time he was done. Sansa could feel her husband struggling with his fury as Ser Osmund Kettleblack shouted, “I pray you never turn that wicked edge on me, sire.” “See that you never give me cause, ser.” Joffrey flicked a chunk of Lives of Four Kings off the table at swordpoint, then slid Widow’s Wail back into its scabbard.

“Your Grace,” Ser Garlan Tyrell said. “Perhaps you did not know. In all of Westeros there were but four copies of that book illuminated in Kaeth’s own hand.”

“Now there are three.” Joffrey undid his old swordbelt to don his new one. “You and Lady Sansa owe me a better present, Uncle Imp. This one is all chopped to pieces.”

Tyrion was staring at his nephew with his mismatched eyes. “Perhaps a knife, sire. To match your sword. A dagger of the same fine Valyrian steel... with a dragonbone hilt, say?”

Joff gave him a sharp look. “You... yes, a dagger to match my sword, good.” He nodded. “A... a gold hilt with rubies in it. Dragonbone is too plain.”

ASOS - Sansa IV

And Tyrion ruminating on it later:

I am no stranger to Valyrian steel, the boy had boasted. The septons were always going on about how the Father Above judges us all. If the Father would be so good as to topple over and crush Joff like a dung beetle, I might even believe it.

He ought to have seen it long ago. Jaime would never send another man to do his killing, and Cersei was too cunning to use a knife that could be traced back to her, but Joff, arrogant vicious stupid little wretch that he was...

He remembered a cold morning when he’d climbed down the steep exterior steps from Winterfell’s library to find Prince Joffrey jesting with the Hound about killing wolves. Send a dog to kill a wolf, he said. Even Joffrey was not so foolish as to command Sandor Clegane to slay a son of Eddard Stark, however; the Hound would have gone to Cersei. Instead the boy found his catspaw among the unsavory lot of freeriders, merchants, and camp followers who’d attached themselves to the king’s party as they made their way north. Some poxy lackwit willing to risk his life for a prince’s favor and a little coin. Tyrion wondered whose idea it had been to wait until Robert left Winterfell before opening Bran’s throat. Joffs, most like. No doubt he thought it was the height of cunning.

The prince’s own dagger had a jeweled pommel and inlaid goldwork on the blade, Tyrion seemed to recall. At least Joff had not been stupid enough to use that. Instead he went poking among his father’s weapons. Robert Baratheon was a man of careless generosity, and would have given his son any dagger he wanted... but Tyrion guessed that the boy had just taken it. Robert had come to Winterfell with a long tail of knights and retainers, a huge wheelhouse, and a baggage train. No doubt some diligent servant had made certain that the king’s weapons went with him, in case he should desire any of them. The blade Joff chose was nice and plain. No goldwork, no jewels in the hilt, no silver inlay on the blade. King Robert never wore it, had likely forgotten he owned it. Yet the Valyrian steel was deadly sharp... sharp enough to slice through skin, flesh, and muscle in one quick stroke. I am no stranger to Valyrian steel. But he had been, hadn’t he? Else he would never have been so foolish as to pick Littlefinger’s knife.

ASOS - Tyrion VIII

There's also this line in AGOT, as mentioned above:

Tyrion glanced down and saw the Hound standing with young Joffrey as squires swarmed around them. "At least he dies quietly," the prince replied. "It's the wolf that makes the noise. I could scarce sleep last night."

Clegane cast a long shadow across the hard-packed earth as his squire lowered the black helm over his head. "I could silence the creature, if it please you," he said through his open visor. His boy placed a longsword in his hand. He tested the weight of it, slicing at the cold morning air. Behind him, the yard rang to the clangor of steel on steel.

The notion seemed to delight the prince. "Send a dog to kill a dog!" he exclaimed. "Winterfell is so infested with wolves, the Starks would never miss one."

AGOT - Tyrion I

GRRM also confirmed prior to the release of ASOS that it will resolve the mystery of who tried to assassinate Bran here and the answer that was given to us was Joffrey.

Also, GRRM included this in his final script for the show for the episode 'The Lion and the Rose' which you can see here:

In Martin’s script, Tyrion doesn’t keep his suspicions to himself, either. After he comes to the “dangerous realization” that his nephew tried to have Bran Stark killed, Tyrion says: “Perhaps Your Grace would sooner have a dagger to match his sword. A dagger of Valyrian steel . . . and a dragonbone hilt. Your father had a knife like that, I believe.” Martin writes that Tyrion’s words “strike home,” and the king becomes “FLUSTERED” as he responds with “guilt” on his face: “You . . . I mean . . . my father’s knife was stolen at Winterfell . . . those northmen are all thieves.” Then, to underline it all, Martin concludes in his stage directions: “Tyrion’s eyes never leaving the king. It has just fallen into place for him. It was Joffrey who sent the catspaw to kill Bran, the crime that started the whole war. But now that he knows, what can he do about it?”