The 20-year-old Italy international, who represents the Nerazzurri, has enjoyed a breakthrough season in Serie A, producing 10 goals and six assists while establishing himself as one of Europe's most highly rated young forwards.
His rapid rise has attracted attention across the continent, with Arsenal, Tottenham Hotspur and Chelsea's ownership group, BlueCo, all tracking his development over the past year.
United's delegation also attended matches at the Pankritio Stadium in Heraklion, where they monitored Greek internationals Christos Tzolis and Christos Mouzakitis.
Both players have been on the club's radar throughout the campaign and remain under consideration as part of a wider shortlist aimed at strengthening multiple areas of the squad.
Esposito's recent international form further strengthened United's conviction. Scouts watched him start and score in successive fixtures against Luxembourg and Greece, performances that reportedly reinforced the belief at Old Trafford that he could develop into a leading centre-forward at the highest level.
Internally, Inter Milan continue to hold the striker in high regard, with club sources praising his progression and the hierarchy viewing him as a key figure for the future.
Despite growing Premier League attention, it remains uncertain whether Esposito would be open to leaving Italy at this stage of his career.
United's attacking plans are also shaped by expected departures, with Dutch forward Joshua Zirkzee increasingly likely to leave Old Trafford.
Inter are understood to admire Zirkzee, creating an indirect link between the two clubs as the off-season window approaches.
Manuel Ugarte has insisted that Uruguay can "always win" at the FIFA World Cup, as he prepares to head to Canada, Mexico and the United States with belief growing inside the squad.
Inter Milan have publicly confirmed their interest in Liverpool midfielder Curtis Jones, with sporting director Piero Ausilio admitting the Serie A giants are closely monitoring the England international's situation at Anfield.
Brazil full-back Wesley has been ruled out of the World Cup because of a thigh injury, with Manchester United-bound midfielder Ederson called up to replace him.
Michael Carrick was backed to continue leading Manchester United after chief executive Omar Berrada said the former midfielder has "earned the right" to stay in charge following a strong spell in the dugout.
