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.
