The Ministers said that they expected May to give a timetable for her successor to be chosen, with June 10 likely to be the start of the official leadership race.
They hope the campaign for the next Conservative Party leader can be completed by the end of July.
May has been under pressure to resign, after a backlash by her own Conservative MPs against her latest Brexit plan.
Since January, Parliament has rejected the withdrawal agreement May negotiated with the European Union (EU) three times for the UK's exit from the block.
May had planned to publish the Withdrawal Agreement Bill on Friday - the legislation required to bring the agreement into UK law - describing it as "one last chance" to deliver Brexit.
However, her proposals - including a customs union arrangement and an offer to give MPs a vote on holding another referendum - angered many party members.
The opposition Labour Party said it was a "rehash" and they would not support the plans.
Andrea Leadsom quit as House of Commons leader on Wednesday evening saying she no longer believed the May-led government's approach would "deliver on the referendum result", the BBC reported.
On Thursday, the Prime Minister met Home Secretary Sajid Javid and Foreign Secretary Jeremy Hunt at Downing Street, where they expressed their concerns about the bill.
More than 12 Conservative MPs are considering running for the position of Prime Minister, with former foreign secretary Boris Johnson currently seen as a front-runner.
On Thursday, the UK voted in the European Parliamentary elections, two months after it was originally due to leave the EU.
Results will not begin to be announced until Sunday night, after voting concludes across the EU.
( With inputs from IANS )