Although the letter of no confidence falls short of the 48 names required to trigger a leadership contest, the UK Parliament's summer break could prove critical for May's future as British prime minister, The Sunday Times reported.
"The numbers change from day to day depending on what's happened but there are about 15 who are fairly consistent in their desire for change. If she has a quiet summer and there are no crises and things are not mismanaged then she might be able to cling on beyond conference, but that is still a big if," a former minister was quoted as saying.
The reports come days after a summer party in the House of Commons last week where May pleaded with her MPs to "go away and have a proper break and come back ready for serious business".
The survey carried out by the Party Members' Project, comes as Parliament stands down for the summer but with bitter party infighting and behind-the-scenes plotting expected to continue over the recess.
It found 21 per cent of members backed Brexit minister David Davis, 17 per cent preferred foreign minister Boris Johnson, while the third choice was backbencher Jacob Rees- Mogg, who has something of a cult following, but was backed by just six per cent of those surveyed.
A quarter of respondents said they did not know or could not say who the next leader would be. However, most Conservatives are keen to avoid a leadership contest until the Brexit negotiations have concluded in 2019, and would prefer to let May finish the job — and carry the can — in the event of a poor deal with the European Union (EU).