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Spurs had hoped to make Adama – who is out of contract in 18 months and has declined a new deal at Molineux – the first signing of the Antonio Conte era and talks between the clubs were progressing until the early part of this week.
But the prospect of a deal is now dead, with Traore instead discussing a loan move to Barca until the end of the season, which is thought to include an option to make the move permanent in the summer for roughly £30m.
Adama began his career with the Catalan giants, progressing through the club's famed La Masia academy and making one appearance in La Liga before a move to English football with Aston Villa in summer 2015.
Conte had wanted to convert the Spain international into a right wing-back, with neither Emerson Royal nor Matt Doherty impressing since the head coach changed systems, but Adama reportedly had concerns about playing the role.
Spurs are continuing talks over a move for Porto forward Luis Diaz, who is valued at more than £40m by the Portuguese club and rated one of the most exciting forwards in Europe.
There is some optimism of doing a deal for the 25-year-old Colombia international, although the clubs remain apart on their valuations.
Meanwhile, Tottenham's hopes of offloading Tanguy Ndombele before the end of the month appear to rest on reaching an agreement with Paris Saint-Germain, with the Frenchman not interested in a move to Everton or Valencia.
As reported by Standard Sport on Wednesday, PSG are working on signing Ndombele on loan until the end of the season but want to downsize Mauricio Pochettino's bulging squad before bringing in new players.
Ndombele has also been linked with Everton and Valencia, with reports in the France saying the La Liga club have an agreement in place with Spurs for the unwanted midfielder.
But Ndombele has no interest in moving to Goodison Park or the Spanish club, with his heart set on a reunion with Pochettino in his homeland.
Spurs are desperate to offload Ndombele, who is not trusted by head coach Antonio Conte, and have lined up a loan move for Fiorentina's Sofyan Amrabat as a potential replacement.