Labour leader Allnatt said he had written to Health Secretary Wes Streeting requesting extra money, but said he had received a “holding response” with no commitment on funding.
The Department of Health and Social Care has been contacted for comment.
The ICB said it did not receive funding to set up new practices and relied on Section 106 cash, which are funds provided by developers, via councils, to mitigate the impact of new house building.
Allnatt said the majority of Section 106 funding went to Leicestershire County Council for road schemes and the building of new schools rather than healthcare.
He said that resulted from a deal done between the borough and county councils before he was elected, and which he hoped would be renegotiated.
Reform UK Leicestershire County Council leader Dan Harrison said the responsibility for prioritising Section 106 money lay “solely” with the borough.
“This is his job, not ours – it’s that simple,” Harrison said.
“If something is our fault, we’ll hold our hands up, but if not, we’ll robustly defend our position.”
In response, Allnatt said: “It is not our decision alone. He’s right in the sense an arrangement had been agreed.
“However, the county council has the whip hand [in deciding how the money is spent].”
The ICB’s chief strategy officer, Pete Burnett, said: “We are committed to continuing to work with Melton Borough Council to explore options for a second primary care site for Melton residents, when guaranteed funding and suitable, affordable premises are identified.
“We are working with the current GP practice in Melton to extend the support it can provide to local residents, which includes a new digital suite at the [practice’s] main site and an approved redevelopment of a property owned by the practice on Sherrard Street to extend clinical services.”
Latham House did not wish to comment.