Several DUP constituency associations have written letters expressing concern at Foster's decision to abstain from a recent Assembly vote on a ban of gay conversion therapy that did not incorporate a specific mention of protections for religious practices.
It is only a small number of the DUP membership who will get to vote in a leadership contest - MLAs and MPs.
Northern Ireland's First Minister Arlene Foster answers questions on her leadership during a visit to the Hammer Youth Centre, in Belfast.
We await the outcome with, erm, interest.
Wells also told the News Letter that party members were unhappy that MLA Diane Dodds attended a recent north-south ministerial meeting, despite the fact that the party is now boycotting such meetings over objections to the Northern Irish protocol.
However, sources later said that the meeting had merely been rescheduled for Thursday.
Foster was the first woman to lead the DUP, a party rooted in the fundamentalist Free Presbyterian Church.
On Tuesday afternoon, she attempted to downplay the scale of the internal revolt, which comes amid mounting discontent among party faithful over her leadership.
"My election as leader of the Democratic Unionist Party broke a glass ceiling, and I am glad inspired other women to enter politics and spurred them on to take up elected office", Foster said.
The move against Foster, who has led the party since 2015, is the latest sign of how Britain's split from the European Union at the end of 2020 has affected the political balance in Northern Ireland, a part of the United Kingdom where some people identify as British and some as Irish.
In a statement, the DUP said its internal democratic electoral processes were a matter for its members and declined to make further public comment on Mrs Foster's future.
The leader of Northern Ireland was fighting Wednesday to keep her job as members of her party tried to oust her over her handling of the fallout from Brexit and other issues.
Mrs Foster was also claimed to have pulled out of a meeting on Tuesday evening with Brandon Lewis, the Northern Ireland Secretary, shortly after reports of the letter began circulating. The DUP is facing anger from the wider loyalist and unionist community for the introduction of an Irish Sea border.
Foster was among only five party members, including fellow Stormont Executive ministers Peter Weir and Mrs Dodds, who abstained.
"The big difficulty is how does this change any of the issues they are unhappy with", one said.
"Sinn Féin will work with all parties to progress social reform, political change and economic prosperity - but we will robustly oppose damaging policies or regressive throwback politics of the past", Sinn Fein's Northern Ireland leader Michelle O'Neill said in a statement.
"The abortion legislation has been imposed from Westminister. But I don't think it's going to solve it".