Survival this season would take Bolton's stay in the top flight to a 10th successive year and guarantee another huge financial pay-day.
Coyle said: "If somebody was to say me 'would you give up the FA Cup to stay in the Premier League?' the answer to that is yes.
"There's no doubt the priority is to stay in this division and it remains our number-one aim.
"But I believe we can stay in the Premier League and progress in the FA Cup.
"Are Tottenham Hotspur the favourites to win tomorrow night? Absolutely.
"Can we go there and win? Of course we can, so we'll go there and look to try and take the game to them in a positive manner."
Coyle could probably do without a trip to the capital, especially as his side face a pivotal clash with relegation rivals Wolves at the Reebok Stadium on Saturday.
Last weekend's dismal 3-0 defeat at Blackburn left Bolton with just one win in eight Premier League matches since Coyle took the helm.
The Trotters currently lie second from bottom with 12 matches remaining and the visit of 16th-placed Wolves has taken on added significance.
With that in mind, Coyle will hand opportunities to a number of fringe players for tomorrow's match.
"We don't have the luxury of having a squad 25 to 30 players - we have a small squad that has been decimated by injuries," said Coyle.
"But we will try to freshen it up as best we can with the Wolves game in mind.
"Yes, the Blackburn game was disappointing, but the only way to bounce back is to come into work, pick yourself up and go again.
"If you feel sorry for yourself then you will never achieve anything.
"The easiest thing in football is to be critical. The coverage of the Premier League is global, so you're always going to come in for criticism.
"So be it. But we've had a meeting with the players about the areas that we can improve on."
Stuart Holden, Jlloyd Samuel, Tamir Cohen, Andy O'Brien, Ali Al Habsi and Mustapha Riga are all in contention to feature, while Ivan Klasnic could start as he continues his recovery from a calf injury.
But on-loan duo Vladimir Weiss and Jack Wilshere are cup-tied and Mark Davies, Chris Basham and Gretar Steinsson could all miss out through injury.
While Coyle prioritises top-flight survival over the FA Cup, Spurs counterpart Harry Redknapp has far loftier ambitions.
"Knowing Harry, he will feel as though he's capable of winning the FA Cup and finishing in the top four of the Premier League," said Coyle.
"He's got the squad to do it and has absolute quality throughout.
"He'll feel that he has an opportunity on both fronts and all credit to him."