Ben Bradley once again urged the Manchester United and England star to meet him so he can explain why he is against providing free school meals for children during half-term.
In a statement to BBC Newsnight, Rashford said: "Growing up we didn't have a lot, but we always had the safety net of the community".
On Friday, Rashford retweeted dozens of offers around the country.
The move follows a busy day on Twitter for Rashford, who has been highlighting the thousands of pubs, restaurants, local councils and other organisations across the United Kingdom who have stepped up and pledged free meals for families in need. That community was my family.
"I'm signing off with a feeling [of] pride tonight".
"We'd like to do our bit for the children in our local community next week", the farm shop's Facebook page says. I am truly overwhelmed by the outpouring of support.
In their Twitter clash earlier this week, Bradley said extending free meals to the school holidays "passes responsibility for feeding kids away from parents to the state".
He added if people wanted "to talk about "superstars" and "celebrities" they should look no further than this list, where he has been sharing messages from businesses offering to help.
Rashford had also clashed with another Tory backbencher, Steve Baker, who voiced his opposition to free school meals during the holidays by saying "not destroying the currency with excessive QE [quantitative easing] is also one of our duties".
From Monday, parents and carers will be able to apply for £15 per week from the council to pay for food during the holiday.
Medway Council and CAM will be providing free school meals.
Cllr Keith Aspden, Liberal Democrat Leader of City of York Council, said: "Whilst it is clear after what happened this week that the government are willing to stand by and do nothing whilst vulnerable children go hungry, as a council we are not".
Fast food giant McDonald's announced on Twitter its partnership with FareShare UK, a charity organisation backed by Rashford, to provide funding for 1 million meals for vulnerable families.
"So, if you are a family or know a family who will be hard it by the Government's decision to let our children go hungry this half term and you live or go to school in Ettingshall then get in touch with me".
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