The Belfast News Letter reported that several DUP constituency associations had written letters expressing concern at her decision to abstain on a recent Assembly vote on a motion calling for a ban on gay conversion therapy that did not incorporate a specific mention of protections for religious practices.
But the fall-out from the UK's European Union exit has put particular pressure on the party top brass as it faces having to weather the storm caused by the Northern Ireland Protocol, which imposes a border down the Irish Sea.
Paraic O'Brien was at Stormont.
The party has argued that it has never supported the protocol and has actively tried to have it overthrown.
The DUP has been under pressure since the Brexit agreement came into force, because it imposes fresh customs barriers between Northern Ireland and the rest of the United Kingdom.
But former DUP adviser Timothy Cairns said a leadership contest was almost inevitable and Foster would be the "greatest political survivor in the history of United Kingdom politics" if she managed to keep her job.
Foster said she will complete work on "a number of important issues for Northern Ireland" in the coming weeks before she steps down from her position.
It is a bold and unprecedented move within the DUP.
It was suggested that under the party rulebook, Mrs Foster could be removed within days because the DUP leader must be re-elected annually before 30 April each year.
She said she would deal with it and move on because she had bigger things to do, including fighting a pandemic.
Poor polling numbers have exacerbated the discontent among the party faithful.
However, while acknowledging that frustration among some DUP figures had been growing for "some time", party insiders on Tuesday questioned the logic behind trying to replace Mrs Foster at such a precarious time.
The DUP said it would not be commenting.
'Whilst understanding that there will be from time-to-time public interest in party processes, these issues, in the first instance, are matters for members of the party and we are not able to make any further comment at this time'.
Mrs Foster was quizzed on the matter during a visit to a youth centre in Belfast on Tuesday afternoon.