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.
Wilshere, 20, was one of five young British players to sign new deals with the Gunners last week, alongside Kieran Gibbs, Aaron Ramsey, Carl Jenkinson and Alex Oxlade-Chamberlain.
Walcott, whose contract expires in the summer, is reportedly being monitored by Liverpool and Manchester City, with the 23-year-old holding out for higher wages and a change in position to striker.
Wilshere told The Sun: "Everybody at the club wants Theo to re-sign. I'm always winding him up, saying 'c'mon Theo, sign'.
"But that is something he has got to sort out. It is up to him and the club to reach an agreement and I hope they do.
"Theo's been great for us and he knows what he means to us and the fans. Five of us British boys have just signed so hopefully he can be the sixth and cap it off."
He added: "We're all great friends, the British boys. We all go out together, have meals together and we all sit together when we're on the road."
Walcott proved his worth to Arsenal on Saturday, winning the penalty which Mikel Arteta scored for the only goal of the game in a 1-0 win at Wigan.