Relegation-threatened Livorno have sacked Domenico Di Carlo after just 88 days in charge and re-appointed Davide Nicola.
Virgil van Dijk claims he knows nothing of interest from English clubs and is keen to prove himself in the Champions League with Celtic.
Barkley has been linked with a £50m move to Manchester United after an eye-catching first season as a Goodison Park regular that has seen him rewarded with three England caps.
Martinez has repeatedly insisted the 20-year-old is not for sale, even claiming a Gareth Bale-style world record bid would not be enough to force the club's hand, and struck a bullish tone again on the eve of the January transfer window.
Asked if he was confident of resisting big-money bids for the likes of Barkley and full-back Leighton Baines - who was a summer target for David Moyes at Old Trafford - Martinez said: "Yes. Absolutely. There is not even a doubt.
"I have said it many times that in football you can't fight the trend of the game but what you can dictate is the timing and everything you do should be for the good of the club.
"We have done it (resist bids) before. Everything that happens needs to be to improve the squad, to give it a longer-term in relation to the strategy of the youth of the players and the blend you have with experience.
"That is based on football decisions and I can guarantee that will be the case in January 100 per cent.
"(Barkley) has a long-term contract. The first half of the season has been a real joy to see him grow and the second half of the season will be even better.
"There is nothing to comment on. The stability we have from a financial point of view allows us to make football decisions and this January window is important for us to make sure we kick on and finish stronger than we started."
Martinez has said previously he would consider allowing Holland international
John Heitinga and Croatia striker Nikica Jelavic to leave in January with the World Cup looming.
Heitinga has been linked with West Ham but the Toffees boss said there was no movement on either player so far.
"We are still focusing on tomorrow's game," he added. "It is such a busy period all that matters is preparing game by game."