The France international left Juventus for Old Trafford in a £93 million deal after an impressive spell in Turin.
Bartomeu told ESPN FC that the Camp Nou outfit had reached an agreement with Juve and Pogba’s agent Mino Raiola a year earlier, which meant they could match any offer that the Italian champions accepted.
However, the staggering fee that the Red Devils were willing to pay – a world record at the time – meant that the Catalan club had to pull out of the race.
"In the summer of 2015, Paul Pogba was playing in Turin and we simply told Juve that, if one day they decide to sell the player, we would be interested," Bartomeu said.
"When they sold the player, they told us what the offer would have to be and we couldn't afford that amount of money at the time.
"So he went to United, and he's making them better as a team because he's one of the stars of the world of football right now."
Pogba has been heavily linked with a move to Real Madrid this summer as Zinedine Zidane’s side plan a summer squad overhaul.
"I prefer not to talk about players from other teams because I am sure there are also players at Barça that can make other teams better,” Bartomeu added.
"What I have to recognise is that United, in the last few years, have created a very good group, good players that are coming through. I am sure next season they will be candidates for the Premier League, and this year for the Champions League."
Barcelona and Manchester United come face-to-face in the Champions League quarter-final first leg at Old Trafford on Wednesday night.
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