The Blues and the Old Trafford outfit have made if clear through third parties that they are prepared to pay up to £25m (€30m) to land the Southampton left-back.
However, Chelsea's bid to capture the left-back is complicated by the fact that he is represented by Stellar Group, the same management firm who represent Ashley Cole.
With Cole out of contract at the end of the season, Stellar will have to balance the interests of England's left-back for the last decade and their emerging star.
As reported by Goal, Manchester United made a £20m move for Shaw on January and the Premier League champions are expected to step up their bid to sign him in the next window.
Sources expect Shaw to join the club that Stellar advise him to, with all signs point to him leaving St Mary's Stadium this summer.
- By Wayne VeyseyMANCHESTER UNITED LAUNCH EVRA CHARM OFFENSIVE
|STAY FOR EVRA
|EVRA'S 2013-14 STATS
The French left-back is out of contract this summer but has an option in his present deal that would allow him to activate a 12-month extension.
Evra was set on calling time on his Old Trafford career and it was revealed this week that Inter are at the front of the queue.
But the 31-year-old has held a series of heart-to-heart talks with boss David Moyes, who has stressed the defender's importance to the camp.
Evra has promised to now make a quick decision. United have also offered a £10,000 (€12,000) a week pay hike on his present £77,000-a-week (€93,000) wages to stay.
- By Paul Clennam