Pressley, who played for the Sky Blues for a year in the mid-1990s, has guided the Scottish League One side to a Scottish FA Cup semi-final against Hibernian on 13 April.
Lee Carsley stepped in as Coventry's caretaker manager in February when Mark Robins left the cash-strapped League One club to take over at Huddersfield Town in the Championship.
Bairns chairman Martin Ritchie is reluctant to lose Pressley, in charge at Falkirk since February 2010, but will not block the 39-year-old's departure.
"We have an agreement with Steven that, should a club come calling, then we wouldn't stand in his way if he felt it was a career progression," Ritchie told STV.
"Steven has been attracting interest from clubs over the last 12 months. So it doesn't surprise me another club is interested in our manager."