The Belgian politician's mandate runs out in November but the European Parliament elections are set for June.
The next European Council president must be elected by a majority of the EU's 27 leaders.
If no successor is found in time, Hungary's prime minister, Viktor Orban, would temporarily hold the presidency.
Hungary will hold the rotating presidency of the Council from July, meaning it chairs the meetings of member state ministers. Nationalist leader Mr Orban would hold the role normally performed by the council president until a replacement for Mr Michel was elected.
Mr Michel, 48, has served as chief of the EU Council, the group of government leaders of the 27 EU member states, since late 2019. Before taking on the EU role he was Belgian prime minister.
"I've decided to stand as a candidate for the European elections in June 2024," he told Belgian media.
"Four years after starting my term as a European leader, it's my responsibility to give an account of my work these past years and to propose a project for Europe's future."
Mr Michel said he would top the list of the liberal Reformist Movement (MR) party, of which he is a former leader, in the European Parliament elections and step down as Council president in July.
Viktor Orban was the only EU leader who met face-to-face with Vladimir Putin last year
There is now pressure on EU heads of government to find a successor before 1 July, when Hungary is due to take over the rotating six-month council presidency.
Mr Orban, Hungary's prime minister, is regarded as Russia's closest ally in Europe, and was the only EU leader who met face-to-face with Vladimir Putin last year.
Last month Mr Orban warned he could "pull the handbrake" on future financial aid to Ukraine, after vetoing a €50bn (£43bn; $55bn) EU package.
In 2022, the EU parliament voted to approve a report that said Hungary cannot be considered a full democracy, accusing Mr Orban of creating an "electoral autocracy".
Mr Michel said announcing his decision in January gave the EU plenty of time to appoint a successor.
But Dutch MEP Sophie in 't Veld accused him of "leaving the ship in the middle of a storm".
"If that is how little committed you are to the fate of the European Union, then how credible are you as a candidate?" she posted on X, formerly known as Twitter.