The 23-year-old goalscoring central defender was a key figure for his national team at the World Cup, finding the net on three occasions - including a last-minute equaliser against England.
Mina joined Barcelona in January 2018 for €11.8m from Brazilian side Palmeiras, but only featured five times in La Liga.
He's seen as a huge prospect for the future by the Catalan club, but they are understood to be willing to let him go out on loan next season to gain experience.
Everton have already expressed an interest, and now reports suggest Pochettino is also keen on the player.
One of the biggest selling points of a move to Tottenham for Mina is that Spurs already have national team partner Davinson Sanchez as part of their defensive unit - who Mina performed brilliantly alongside during Colombia's World Cup campaign.
However, Spurs could reportedly be put off a loan deal for Mina given that Barcelona would insist the player played a certain number of games or minutes during his loan spell.
Pochettino enjoys healthy competition within his Spurs side, with very few positions truly locked down by players. Even Toby Alderweireld, a key player for Tottenham, struggled to earn a place back in the side after returning from injury.
As such, obligations to play a certain player over others, especially in a position like centre back, where competition for places is currently open and healthy, would likely go against Pochettino's style of management.
If Pochettino really fancies acquiring the services of Mina and wants Spurs to force the issue, then the London club might be forced to pay the Colombian's release clause at Barcelona -which currently stands at £88m.
Spurs are already looking to sign Colombian international teammate Wilmar Barrios from Boca Juniors.