Royle wants the Danish international as cover for two areas - midfield and centre-back - exposed by the current Goodison injury crisis. Thomsen had an impressive Euro 96 campaign.
Everton and Ipswich have been discussing a fee for the past couple of days, but the offer is close to being accepted.
The Brighton manager, Steve Gritt, has made it clear he has no wish to get caught up in off-the-field distractions as talks continue in a bid to resolve the club's future.
"The football side is what concerns me. I want to concentrate as much as possible on the football and want this to be the case for the players."
Gritt, who took over at the the Goldstone Ground five weeks ago, said his top priority is to do everything possible to keep Brighton in the Football League and all his efforts are concentrated on team matters.
"I'm sure if and when off-field matters become sorted out and settled, and if we can also make sure the job we've done is right, then hopefully Brighton will still be a Third Division side next season," he said.
More talks between the Brighton chairman, Bill Archer, and Dick Knight, the head of the consortium trying to take over the club, will take place this week.
The discussions have been organised by the Football Association and are taking place under the mediation of the Centre for Dispute Resolution.Reuse content