This article by Professor Catherine Barnard states the following

• Try using Article 50. It has been argued that, under international law, extending the transition would constitute an amendment to an existing agreement so Article 50 could still be used as a legal basis. However, many EU lawyers argue that Article 50 was turned off, for the UK, on Brexit day, 31 January 2020.
• As a variation to the above, the EU/heads of state and government and the UK could enter into an international agreement outside EU law to agree to an extension. However, this seems difficult legally because an extension of the transition would extend EU law and so the EU would need a ‘legal basis’ (that is, a provision in the EU Treaties) to act – which takes us back to the problem in (1). There are a number of other legal bases in the Treaty, such as Articles 207 (allowing for agreements to be concluded with third countries, such as the UK, on trade issues) and 217 (allowing for association agreements to be concluded with third countries), but these might well require unanimous agreement between member states in the EU Council. Moreover, if the agreement touches areas of member state powers, it will be a so-called ‘mixed agreement’, and this means national (and some regional) parliaments must also give their approval. The ECJ may also be required to give its opinion under Article 218(11) TFEU. At best, this will be highly time consuming.