The 25-year-old is closing in on 300 appearances for Atletico in all competitions and has a contract at the Wanda Metropolitano until the summer of 2026.
A recent report claimed that Man United and Man City were both weighing up summer moves, having been given encouragement in their pursuit of the Spain international.
However, according to reports in Spain, Barca would also be in the hunt for the midfielder if it became clear that Atletico were prepared to sell for the right price, which could be as much as €150m (£138.9m).
Saul has contributed four goals in 37 appearances at club level this season, including one against Liverpool in the first leg of their Champions League last-16 clash.
The Spaniard operated as a central defender earlier in his career but largely features as a midfielder for Diego Simeone's side, showing his versatility by operating in the centre and on both wings. body check tags ::