Several backbenchers broke cover over the weekend to criticise her lack of action on domestic issues.
Intensifying Tory infighting will not go unnoticed in Brussels, where the EU General Affairs Council is expected on Monday to approve guidelines for chief negotiator Michel Barnier to follow during talks on a post-Brexit transition period.
There are concerns they will produce a Tory meltdown in London.
A former cabinet minister said: "It looks like these elections are going to be very bad".
Eurosceptic former Cabinet minister Theresa Villiers said she was anxious about a "dilution of Brexit" while Remainer Heidi Allen told the PM to "get a grip" because the Tories are "letting this country down", and former minister Rob Halfon appeared to liken Mrs May to a "tortoise".
"The soundings off have to stop", she said.
Mr Timothy said: "I think the party needs to set up processes that will enable it to debate its future objective and policies because if that is left until 2020/21, that is going to feel too late in the electoral cycle".
"I think she is doing an important job for our country".
During the reshuffle earlier this month, Mrs May had wanted to move Mrs Greening to the Department for Work and Pensions, but she refused and quit the government instead.
Rees-Mogg also branded Philip Hammond "problematic", likening the tension between the Chancellor and Prime Minister to the disagreements between Tony Blair and Gordon Brown under New Labour.
Yet Ms Greening's candour is pertinent for three other reasons.
The £3,387 grants towards living costs were available to students from families with annual incomes of £25,000.
Mr Walker told World at One: "It seems to happen every two weeks, we have these spasms within the party where there is this rocking of the boat, there is this off the record briefing of the media, and I simply can't understand, really, what is motivating this small minority of colleagues to behave in this way".
Pro-Brexit campaigner and former minister Theresa Villiers said Britain appeared to be heading for a "dilution of Brexit", with a deal that would "keep us in the European Union in all but name".
The energy minister was responding to a colleague who said he was "getting some shit from the usual subjects about sell-outs and traitors" over the £39bn "divorce bill", telling him: "The "sell out traitor mob" should be ignored".
'I would hypothesise that they are mostly elderly retired men who do not have mortgages, school-aged children or caring responsibilities so they represent the swivel-eyed few not the many we represent'.
The environment secretary Michael Gove's supporters were also limbering up for another push, he added.