British Conservative Rishi Sunak on Sunday announced he is standing to be prime minister, just weeks after failing in a first attempt and setting up a potentially bruising battle with his former boss Boris Johnson.

Ex-finance minister Sunak said he had a "track record of delivery" as he vowed to lead Britain out of a period of "profound economic crisis", which experts say has been worsened by the aborted policies of outgoing leader Liz Truss.

"I want to fix our economy, unite our party and deliver for our country," he said in a short statement posted on Twitter confirming his widely expected candidacy.

The no-frills announcement contrasted with his last failed bid to be Tory leader, when he faced criticism for a slickly produced video launched just days after he had helped depose ex-prime minister Johnson by resigning in July.

Sunak is the second Conservative MP to declare a run at the top job after cabinet member Penny Mordaunt launched her campaign on Friday.

Johnson, 58, is also anticipated to enter the potentially week-long contest, after he cut short a luxury Caribbean holiday to return to Britain on Saturday and attempt an audacious political comeback less than two months after leaving office.

He is said to be intensely lobbying Conservative colleagues ahead of a Monday deadline to secure the 100 nominations required to face a vote of Tory MPs.

Sunak has raced ahead in that count, with the 42-year-old crossing the minimum threshold on Friday and currently boasting the public backing of 129 Tory lawmakers, according to a BBC tally.

That compares to Johnson's 53 and 23 for Mordaunt, though the ex-leader's allies insist he has already amassed the 100 nominations.

"We've got the numbers," Northern Ireland Secretary Chris Heaton-Harris reiterated on Sunday, telling Sky News the ex-premier was "keen to see what the parliamentary party thinks".

The Tories have been forced into this second, now expedited, leadership contest since the summer following Truss's resignation after only six weeks into her tenure over her calamitous tax-slashing mini-budget.

If two candidates remain after Tory MPs vote on Monday, the Conservatives' approximately 170,000 members will get to make their choice next week, with Johnson still seen as the favourite of the grassroots.

The divisive Brexit architect only relinquished power in early September, two months after announcing his resignation following a government revolt over a slew of scandals.

Amid growing calls for unity within the fractured ruling party, he and Sunak held talks running late into the night, reports said on Sunday, as the feuding political heavyweights battle behind the scenes.

Few details have emerged about what The Sun dubbed a "secret summit". The Sunday Times said it began at 8pm and was ongoing at 11pm, "in the hope of burying the hatchet and thrashing out a deal to govern together".

"We're all waiting for white smoke," an ally of one told the newspaper.

However, Sunak's announcement on Sunday suggests he is confident he can vanquish Johnson.

But Mordaunt, who missed out on the last contest's run-off by just eight MPs' votes, is adamant she alone is able to bring the party together.

"I'm best placed to unite our party," the 49-year-old told the BBC, arguing she was "the halfway house" between Sunak and Truss in the summer race. (AFP)