Labour’s deputy leader Tom Watson demanded that the ad watchdog investigate, calling the allegation “at best misleading and at worst a complete myth”.
He wrote to the ASA on Monday, objecting to a graphic put out by the prime minister’s official Twitter account.
It said that the NHS would be funded by a “Brexit dividend” – highlighted by large, bold letters – and “the country contributing a bit more”.
On Sunday, the prime minister unveiled a plan to give the health service a funding boost of £20bn, an average of 3.4% every year for the next five years.
She referenced the Vote Leave campaign’s pledge to spend Britain’s EU contribution on the NHS and delayed details on tax rises until November.
Questions were soon raised over the announcement, with the Institute for Fiscal Studies stating “there is no Brexit dividend”.
Other Tory MPs have dismissed her claim as “tosh”.
In a letter to the ASA’s chief executive Guy Parker, Mr Watson quotes the watchdog’s own guidelines.
They state that advertisements put out by the government or local authorities must not be misleading and that: “Object claims must be supported by robust evidence”.
“Local authorities, government and other public bodies should be aware that their ad campaigns, like those of commercial companies, are subject to the rules in the advertising codes.”
The guidelines add: “The ASA will intervene if an ad by a public body crosses the line, whether that’s through being harmful, offensive or misleading.”
Mr Watson’s letter quotes the IFS, which said on Sunday: “From an arithmetic point of view, from this period, there’s no money.”
A spokesperson for the ASA said the complaint was currently being assessed.
A decision will be made on whether to launch an investigation.
Sky News contacted Number 10 for a response. A spokesman repeated the prime minister’s words over the weekend.