The 18-year-old Saints left back remains the top target for both clubs, but Mauricio Pochettino is desperate to keep hold of England’s latest debutant.
It is still unclear as to whether United or Chelsea will be able to lure Shaw away from St Mary’s this summer, so both have been busy weighing up alternatives.
21-year-old Spaniard Bernat has impressed in La Liga this season, and is capable of playing at either left back or left midfield.
Bernat is a product of the Valencia youth academy, and has gone on to rack up 84 first team appearances since breaking into the first team in 2011.
His future has been the topic of speculation however, as he is out of contract at the Mestella in a year, and is yet to agree an extension.
He currently has a £16.7million release clause in his contract, but both clubs would be confident of signing him for less with his contract nearing its end.
Reports suggest United and Chelsea may face a fight for his signature with Inter Milan, if they fail in their pursuit of Manchester United left back Patrice Evra.