Sean Dyche has been a long term admirer of McArthur and Burnley had a £4 million offer rejected just over a week ago.
Speaking at the recent fans' forum event Latics manager Uwe Rosler revealed this had been rejected and that there was no needto sell McArthur.
The Clarets have since come back with a renewed offer of £5 million for the Latics to consider.
McArthur is an integral part of the Latics squad and has proven pedigree at the top level where he formed a superb central midfield partnership with James McCarthy.
With the Latics struggling on the recruitment front the loss of McArthur would be damaging to the promotion chasing campaign.