French President Emmanuel Macron has rejected a resignation offer from the prime minister he appointed, saying instead that the government must "stay on task and act".

Elisabeth Borne was criticised by some commentators after Mr Macron's coalition lost its majority on Sunday.

Her future looked under threat after the result, which left the president needing to secure support from rivals.

He is set to meet his political opponents for rare talks on Tuesday.

But neither Marine Le Pen's far-right nor Jean-Luc Mélenchon's left-green alliance are keen to work with Mr Macron's centrist government, which is 44 seats shy of a majority and desperately trying to avoid political paralysis.

The Élysée palace said Ms Borne formally offered to resign on Tuesday morning, submitting a letter to Mr Macron - the head of state - who turned it down.

Mr Macron refused the offer "so that the government can remain on task and act", the Élysée palace said, adding that he would seek "constructive solutions" to ending the deadlock that threatens his agenda during his second term.

It is customary for the French prime minister to offer to resign following parliamentary elections. Often, the president will simply reappoint the same person so that he or she can begin constructing a new government.

But the situation is different this time, as Mr Macron is telling Ms Borne to stay in office with the same cabinet without resigning.

It is probably an effort to buy time, analysts suggest, adding that at some point, a new government will be formed which may or may not have Ms Borne at its head.

Meanwhile, party representatives will visit the Élysée Palace for high-level talks on Tuesday and Wednesday.

Analysts say the president may be eyeing a deal with the right-wing Republicans. The party confirmed that its leader, Christian Jacob, would attend the talks.

But prospects of a deal appeared dim in the run-up to the talks, as Mr Jacob hit out at Mr Macron in an interview with France Inter radio. "He's the one who's been arrogant and now he calls for help," he said.

"We are very clear on our stance, we are in the opposition to Emmanuel Macron and will remain there," he added.

Ms Le Pen will take part but Mr Mélenchon will not, the AFP news agency reports.

Socialist Party leader Olivier Faure and Communist Party boss Fabien Roussel, members of the Nupes left-wing alliance, will also meet Mr Macron.

Having no clear majority in the National Assembly is likely to make it far harder for Mr Macron to implement his agenda during his second five-year term.

He has laid out a series of plans to tackle the spiralling cost of living, including food vouchers and enhanced benefits. Another big reform is gradually raising the retirement age from 62 to 65, which has proved unpopular with much of the electorate.