Sky sources understand Burnley have made four bids for the Scotland international, all of which have been rejected, with the last valued at just below £4million.
Leicester became the second Premier League club to have a bid turned down for the player on Tuesday, but it is understood Wigan chairman Dave Whelan will allow the 26-year-old to leave if his valuation is met.
Nigel Pearson has turned to McArthur after becoming frustrated in his pursuit of former Real Madrid and Inter Milan midfielder Esteban Cambiasso.
The Leicester manager spoke earlier this week about his desire not to be 'held to ransom' as the transfer deadline approaches so it remains to be seen if the Foxes will return with an improved offer for McArthur.
"We've had our fingers burnt in the past with doing deals late in the window,” Pearson said.
“I recognise we need one or two extra players but I'm certainly not going to be held to ransom.
“My advice to the football club is if we can't get the people we want, I don't want to pad it out with players who are not the right fit or not good enough.
"There are more hoops to jump through this year but I'm not involved at that level. I'm very happy with the players here - the numbers are the issue.
“We have an interest in a midfield player and if we can add another defender and a forward, I'd be happy."