Mr Juncker is due to stand down at the end of October 2019, following a five-year term.
But the European Commission President is said to be hoping to stay on for longer if there is chaos following next May’s European Parliament elections.
Mr Juncker reportedly warned EU leaders that an inconclusive result in the European Parliament elections would make selecting his successor more difficult.
A source said to the Sunday Times: “Some commissioners reckon they’ll need an extra two or three months, but Juncker would quite fancy an extra year in the job.”
The European Commission’s president is chosen by EU leaders but must be approved by Europe’s Parliament.
But MEP elections in May 2019 could plunge European Parliament into turmoil as Europe’s two main centrist parties – the European People’s Party and the Socialists & Democrats – are predicted to lose their majority.
Far-right and Green parties are expected to make major gains.
It comes as Theresa May battles to secure a Brexit deal with the EU ahead of the UK’s departure from the bloc in March 2019.
The Prime Minister is facing major divisions within her party over her Brexit blueprint as well as reports of opposition from the EU.
Speaking today, Mr Juncker said slow progress is being made towards a deal.
Mr Juncker said: “I have the impression that we are moving slowly but surely towards a definitive Brexit deal which should be concluded in the weeks to come.”
But Mrs May was dealt her latest blow as Tory Brexiteers and the DUP pledged they would vote against her Brexit deal if it threatened to break up the UK.
Writing in the Sunday Telegraph, deputy chair of the powerful European Reseach Group Steve Baker and DUP Brexit spokesman Sammy Wilson said: “We share the Prime Minister’s ambition for an EU free trade agreement, but not at any price, and certainly not at the price of our union.
“If the Government makes the historic mistake of prioritising placating the EU over establishing an independent and whole UK, then, regrettably, we must vote against the deal.”
It follows the shock resignation of transport minister Jo Johnson, with speculation high that other ministers could follow Boris Johnson’s brother.