Burnham is due to step down as a Leigh MP next year to fight for the mayoralty of Greater Manchester, and Nuttall is considering a battle to steal the seat from Labour.
A by-election in the seat would be the first major test of Nuttall's plans for the party, and in particular the ability to win over Labour voters in the north.
While both Liverpool and Manchester actually voted to remain in the EU in last summer's Brexit referendum, many of the surrounding areas backed a Leave vote.
And overall the North-West backed Brexit by 59 per cent to 41 per cent.
Success would represent the first time Ukip has won a seat at a by-election – current MP Douglas Carswell and former MP Mark Reckless both defected to the party from the Conservatives before retaining their seats in the contests that followed.
Former Conservative Bob Spink also briefly served as a Ukip MP for the seat of Castle Point between 2008 and 2010, when he ran as an independent candidate, when the Tories regained the seat.
While Burnham has not announced a date for his departure, Manchester will vote for its new mayor on 4 May 2017.
Liverpool Walton MP Steve Rotheram, an aide to Labour leader Jeremy Corbyn, will be fighting to be selected for the Merseyside mayoralty on the same day, meaning two Commons seats in the North West could be up for grabs before the middle of next year.
Voters in the West Midlands will also choose their new regional mayor in early May, although none of the declare candidates – which include John Lewis managing director Andy Street – are currently sitting MPs.
A spokesman for Nuttall declined to comment.
Separately, Ukip is also contesting the Conservative seat of Sleaford and North Hykeham, following the resignation of Brexiteer MP Stephen Phillips.
The Lincolnshire constituency, which registered a 56 per cent vote for Phillips and the Tories last year, will choose its new MP on 8 December.