Matt Hancock, the health secretary, told BBC Breakfast on Monday morning there “has been communication” between the footballer and the prime minister.
But the Manchester United and England star said he had not spoken to Johnson since June.
Thousands of free meals will be provided to children by businesses, local authorities and community groups on the first day of half term, following a furious backlash against the government.
Dozens of people from a range of organisations, including Conservative-led councils, have stepped in to help.
A petition from Rashford, who has been spearheading demands for free meals to be extended in England over the school holidays, has passed 800,000 signatures.
Last week the government voted against a motion tabled by Labour in the Commons to extend the free school meals programme as called for by Rashford.
Labour leader Keir Starmer said on Sunday it was “not too late to do the right thing” and confirmed his party would force another vote if ministers did not back down in time for the Christmas holidays.
Tim Loughton, a former children’s minister, was among the Conservative MPs to warn Johnson over the weekend it had been a “mistake” not to continue with free meals for hungry children.
“Free school meals is just one of those totemic things – it is like the NHS, it can do no wrong,” he told BBC Radio 4’s The World This Weekend.
Speaking to BBC Radio 4′s Today programme on Monday morning, Hancock hinted that further help could be given and that the government could stage a partial climbdown in time for Christmas.
“Our attitude and our purpose it to ensure that everybody gets the support they need and no child, of course, no child should go hungry, nobody could possibly want that,” he told BBC Radio 4’s Today. “The question is how best to do it.”