Two groups of Tory MPs have launched campaign groups aimed at shaping the future direction of their party.
Former cabinet minister Esther McVey has launched Blue Collar Conservatism, aiming to target "working people".
Work and Pensions Secretary Amber Rudd is among those backing One Nation Conservative Caucus, a group opposing a no-deal Brexit.
The contest to replace Theresa May has - unofficially - begun, with several figures already saying they will run.
The prime minister has promised to set a timetable for the election of her successor after the next vote on her Brexit plan.
MPs are due to vote on the Withdrawal Agreement Bill - required to put the PM's deal into UK law - in the first week of June.
Ms McVey, who quit as work and pensions secretary last November in protest at Mrs May's deal, has already announced she will stand in the forthcoming leadership contest.
Launching the Blue Collar Conservatism group at an event in Westminster, she said the next leader of the party must be a "Brexiteer who believes in Brexit".
She added that the Tories' "failure to deliver Brexit" meant the party had failed to capitalise on winning over voters who abandoned Labour at this month's local elections.
"A majority of these voters voted to leave the EU, and on this we have broken their trust," she said.
The Leave-supporting MP added it was "essential" for the UK to leave the EU before the new deadline of 31 October, with or without a deal.
She also called for international aid spending to be lowered to 2010 levels, which she said would free up £7bn in extra funding for schools and the police.
Doing so would allow the Conservatives to "match people's needs and priorities", she added.
According to the Mail on Sunday, Ms McVey will embark on a "pub tour" campaign, where she will also call for the HS2 rail project to be scrapped.
Meanwhile, the One Nation Conservative Caucus has said it expects to be involved in the debate over the party's future and have a say in whoever wins the Tory leadership race.
The group is reportedly aiming to block any candidate who backs a no-deal Brexit.
Analysis by Ben Wright, BBC political correspondent
The unofficial contest to replace Theresa May is already up and running, with candidates declaring their desire for the job and policy platforms being set out.
Work and Pensions Secretary Amber Rudd has not - yet - thrown her hat in the ring but spearheads a 60-strong block of Tories called the One Nation Caucus.
On Monday, it will launch a declaration of values that seeks to cement the post-Theresa May Tory Party in the centre ground.
Sources deny it's a "stop Boris Johnson" effort, but the group is emphatic there must be no future coalition between the Conservatives and The Brexit Party and is stressing the importance of issues like the environment.
Amber Rudd, who spoke at an event to launch the group, has said it is "entirely possible" she will launch a bid for the leadership once Mrs May steps down.
Speaking at the launch she said: "The Conservative Party is entering a new phase and we here in this room are determined to shape that phase."
Ms Rudd added: "Sometimes our voices aren't heard quite as vocally as they should be."
"Part of the launch today is to say we are going to be stepping up, making ourselves heard because we are proud and honest and strong about what we believe in."
The group also includes MPs such as International Development Secretary Rory Stewart, who has confirmed he will put himself forward to be next Tory leader.
Former cabinet ministers Nicky Morgan and Damian Green, as well as backbencher Sir Nicholas Soames, also made speeches at the event.
Mr Green told the BBC the group would hold "hustings" to find out how leadership candidates would seek to find a way out of the current Brexit deadlock.
"One of the things that we all agree on is that it would be massively better for this country to have a deal, so we don't see no deal as a good option for this country", he said.
He added that the group would promote internationalism, the environment, as well as protecting consumers from corporations and an "over-mighty state".
"Obviously at some stage over the next few months we will have a new leader as well, so we want to make sure that the leader ascribes to these values", he added.
Former Foreign Secretary Boris Johnson has also said he will run for the leadership, telling an event last week: "I don't think that is any particular secret to anybody".
Former Brexit Secretary Dominic Raab, Health Secretary Matt Hancock and Chief Secretary to the Treasury Liz Truss have also spoken also at a Daily Telegraph event on the future of the Conservative Party. All three have been tipped as contenders.
Ms Truss said her party needed to be bolder on issues such as housing while Mr Raab called for cuts to income tax.
In a Conservative leadership contest, MPs hold a series of ballots, with the candidate gaining Tory MPs launch rival campaign groups the fewest votes eliminated at each stage.
Once the field is reduced to two, the winner is chosen by a vote of party members.