The 20-year-old was keen to force an exit from Elland Road last summer following the club's relegation from the Premier League, but a transfer ultimately failed to materialise.
Gnonto had previously made a notable impression in the Leeds first team following his arrival from FC Zurich in June 2022, contributing with four goals and four assists in 28 appearances across all competitions last season.
The Italy international has since struggled to make his mark under new manager Daniel Farke, as he has only chipped in with one goal and one assist in seven starts and 13 substitute outings in the Championship this term.
Nevertheless, Leeds have relied on other attacking stars to help them climb up to fourth in the second-tier table, with Crysencio Summerville, Joel Piroe, Daniel James and Georginio Rutter all outperforming Gnonto in the first half of the campaign.
Gnonto is still widely regarded as a bright prospect for the future, though, and has been tipped to make a swift return to the Premier League.
The likes of Arsenal, Manchester City and Crystal Palace were all credited with an interest in Gnonto when he was plying his trade in the top flight with Leeds, while Everton have since been linked with the attacker.
However, according to Sportslens journalist Dean Jones, West Ham are "primed to make an enquiry" for the youngster in the "coming days".
Jones adds that the Hammers considered moving for Gnonto last summer when Leeds were reluctant to cash in on the attacker, and they will look to test the waters at some stage this year.
Fellow Premier League side Fulham are also said to be keeping tabs on Gnonto, who is under contract at Elland Road until the summer of 2027.
Leeds allegedly paid just £4m to sign Gnonto 18 months ago, and they now value the former Inter Milan youngster in the region of £30m.
Gnonto is not the only attack-minded player said to be on West Ham's radar, as a recent report claims that David Moyes's side have identified Ajax winger Steven Bergwijn as a potential target.
The Hammers are believed to be considering the addition of a new attacker this month as they are having to cope without Mohammed Kudus, who is representing Ghana at the Africa Cup of Nations, while Michail Antonio has suffered a setback in his recovery from a knee injury.