There has been constant speculation of late linking 17-year-old Shaw with a summer move to the European champions at the end of this season following a breakthrough campaign at St Mary's in which he has been a revelation since former boss Nigel Adkins pitched him into the side in November.
However, it now seems the Londoners are more interested in taking 21-year-old Clyne to Stamford Bridge in a £7.5million swoop when the transfer window re-opens.
Clyne has impressed in his first season in the Premier League with 26 starts from their 30 matches following a £2.5million switch from Crystal Palace last summer.
He is also a regular member of Stuart Pearce's England Under-21 squad, playing 74 minutes of the friendly win against Romania on Thursday.
Meanwhile, Shaw used a Twitter Q&A at the weekend to seemingly push for a move to Stamford Bridge this summer.
When asked who is favourite club was, apart from Southampton, Shaw first replied: "I think most people know that." He then answered the question again by saying: "Chelsea."