England fly-half Wilkinson, 28, is out of contract with the Falcons at the end of this season.
And speculation that Wilkinson could head elsewhere next year intensified after the club and his mentor Steve Black parted company earlier this week.
Black, who has worked closely with Wilkinson for several years, left Kingston Park by mutual consent.
Wilkinson has yet to agree fresh contract terms on Tyneside, although it is understood negotiations between the club, player and his advisers are under way.
Falcons rugby director Fletcher said: "Jonny is obviously out of contract in the summer, as are about 12 or 13 of our other players, which is the case at this time every year.
"When that happens, the player and his management team have to make a decision about what they feel is best, and it is not any more complicated than that.
"There is always speculation about Jonny.
"He has been here for 10 years, and there has been speculation for each of those 10, but yet he's still been here for all 10.
"Jonny and 'Blackie' have a very close relationship, both professionally and personally.
"You could say that might be a factor in a decision, but then there are any number of factors and lots of things to consider.
"It is no different to any other player, you do what you feel is best, but what I would add is that players very rarely choose to leave this club."
Premiership heavyweights Leicester and Wasps would probably spearhead domestic interest in England's record international points scorer and 2003 World Cup winner.
But lucrative overseas offers could also be expected for English rugby's hottest property.
Wilkinson, meanwhile, will start his third game since the World Cup final in Paris when Newcastle host European Challenge Cup opponents Connacht on Sunday.