According to Daily Express, the Netherlands centre-back has emerged as a favourite of Louis Van Gaal's for his country, and the new Manchester United boss even gave de Vrij the captain's armband during the recent friendly against Ecuador.
De Vrij received the captaincy after United striker Robin van Persie was withdrawn in the second half, but the two could be linking up at Old Trafford next season too.
"It was an enormous honour," de Vrij said regarding the captaincy. "After Robin was changed the technical staff pointed to me - it was fantastic, really something to be proud of."
Reports in Italy still claim that Lazio remain the favourites to sign the Dutchman this summer, but a number of Premier League teams - including United, Liverpool, Arsenal, Newcastle and Everton - have been watching him closely.
Still United will have to move fast if they want to beat the Italians, and the Daily Express reports that Van Gaal could be preparing an early bid to steal a march.
Liverpool boss Brendan Rodgers wants reinforcements this summer ahead of a Champions League campaign, but United also have money to strengthen and need replacements for Nemanja Vidic and Rio Ferdinand.