Bayern are still in discussions with Ajax over the transfer of Dest with the two clubs trying to find a compromise on transfer fee.
The chase is heating up for the 19-year-old, with both Barca and Bayern keen on bolstering their defence with the young Ajax full-back.
Bayern Munich were leading the race late last week and were the only club to make a direct approach to Ajax. However, Ronald Koeman's side have been long-standing admirers of Dest but any interest was dependent on them selling players.
With Nelson Semedo closing in on a move to Wolves, Barcelona are now looking for a new right-back with Dest and Max Aarons on their radar. Dest is understood to be Barcelona and Koeman's preferred option but Aarons has been a long-term target and sources close to Aarons say Barca began sounding him out about a move in May but have had a loan bid turned down.
Barcelona's season starts on Sunday and only have Sergi Roberto as an option at right-back so will look to move quickly this week once the money comes in from Semedo.
For Bayern Munich, they are looking for another right-back to slot in alongside Benjamin Pavard and have long held an interest in Dest. The two clubs are understood to be in talks but there's been no breakthrough on transfer fee.
Tom Hamilton, Moises Llorens and Sam Marsden contributed to this report.