Chelsea have endured a mixed transfer window despite Conte managing to bring in four new recruits.
Deals for Antonio Rudiger and Willy Caballero were completed quickly, while Tiemoue Bakayoko and Alvaro Morata have also moved to Stamford Bridge.
But, despite their activity, Antonio Conte wants more signings if they are to become the first team to retain the Premier League since Manchester United in 2009.
The Blues boss is said to want Southampton defender Virgil van Dijk, though will face competition from Jurgen Klopp's Liverpool.
And The Sun say Conte also wants Abramovich to pursue a deal for Tottenham's Rose.
The defender has been out of action with a knee injury since January but is expected to be fully fit by the end of this month.
And Conte has pinpointed Rose as a priority signing.
The England international has been the club's first-choice full-back since Mauricio Pochettino succeeded Tim Sherwood in 2014.
Rose has also been linked with a move to Jose Mourinho's Manchester United, as well as Serie A giants Inter Milan.
But Conte is an admirer and, with their bid for Juventus' Alex Sandro unlikely to be accepted, the Tottenham defender is on their radar.
The report adds that, should Spurs manage to retain the 27-year-old, then Chelsea could try and re-sign Ryan Bertrand from Southampton.
Bertrand left the Blues three years ago but has impressed at St Marys and might be set for a shock Stamford Bridge return.
Source: Daily Express