The Sunday Mirror says Spurs may be able to sign left-back Masina, 23, for a knockdown price as he has only 18 months remaining on his Bologna contract.
It follows reports that Rose could be the subject of a £40m ($53m) bid from United in January, with Luke Shaw potentially moving in the other direction as a makeweight.
He made his first appearance for Tottenham since suffering a serious knee injury in February in the 1-1 away draw against Real Madrid in the Champions League last month, but was left out of the matchday squads for the derby defeat against Arsenal and the home draw against West Brom on Saturday, 25 November.
The England international, who was fined two weeks' wages for his interview with The Sun, insisted earlier this month that the controversy was behind him and that he continued to get along well with manager Mauricio Pochettino.
"For my part, it is forgotten," Rose was quoted as saying by ESPN. "I think it is forgotten on the club's part as well. There is no point dwelling in the past if we want to take the club forward and go that one stage further and win the league and go as far as possible in the Champions League. On my part, it is all forgotten about and I am just concentrating on the job in hand – helping Tottenham.
"Me and the gaffer are fine. We speak most days. The last three years, the lads will tell you I am the one in his office the most, speaking to him and going through videos and sharing text messages. Even going round his house," the 27-year-old added.