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.
Rooney is inside the final 18 months of his contract and was the subject of a failed Chelsea bid back in pre-season.
Mourinho now holds out little hope of landing last summer's number one target, but on the eve of United's trip to Stamford Bridge he suggested United could sell to a "non-domestic rival" in the coming months.
Mourinho told national newspapers: "Man United made it clear in the summer they don't swap or sell players to their direct rivals.
"They were clear in the way they approached the Rooney situation. For them, it's pretty clear. They don't swap. They don't sell.
"I think maybe this summer they will try to sell to a non-direct rival. But they were very clear."
Moyes does not feel the Chelsea manager is correct in his assessment that Rooney could be on his way to an overseas club, but he refused to resort to similar tactics regarding Mourinho's players, including the unsettled Juan Mata.
"I don't talk about players at other clubs. It's not my style," the Scot said.
"Only time will tell and give you the answer (about Rooney). I completely disagree. I can only tell you I disagree.
"The club do the right things, there is no doubt about that. We'll make sure we do the things we need to do."
Rooney is approaching the 10th anniversary of his move to United. He joined from Everton in August 2004 and is the fourth highest goalscorer in United's history.
A groin injury means he will be absent from the line-up for Sunday's match at Stamford Bridge.
Moyes, who succeeded Sir Alex Ferguson last summer, has revealed he received an apology from Chelsea over comments Mourinho made about the Red Devils.
Stories appeared this week in which Mourinho indicated he had been made aware by someone within the United camp that all was not well at Old Trafford.
Speaking in the Evening Standard, the Portuguese had said: "My feeling, which is based on years of communicating with Sir Alex and some inside information, is Man United are not happy."
But ahead of his side's clash against Chelsea at Stamford Bridge on Sunday, Moyes stated the Blues have apologised.
"I've seen it and had an apology from him and his department," Moyes said.