“You know in the House, when we don’t have the votes to pass something, we work on the bill until we pick up the votes to pass it. That includes GOP votes - & yes, GOP have voted for my leg too,” she tweeted to the Texas senator.
“The Senate should try it sometime! People are going hungry as you tweet from vacay.”
Earlier Ms Ocasio-Cortez had taken a shot at majority leader Mitch McConnell for letting senators leave Washington DC.
"People across the country are going hungry, COVID is set to explode, and Mitch McConnell dismissed the Senate last week," she tweeted.
"I don’t know how these people can sleep at night. I really don’t.”
Washington has failed to pass any virus relief since the bipartisan $2.2 trillion CARES Act in March.
Mr Cruz then fired back by accusing Democrats of blocking a Republican-backed pandemic relief package.
"Why is your party filibustering $500 billion in COVID relief?" asked Mr Cruz.
"And Joe Biden is cheering them on. Thinking that blocking relief somehow helps Dems win Georgia.”
Ms Ocasio-Cortez shot back that the Democrat-controlled Congress had passed Covid relief in May and sent it to the Senate.
"AOC seems not to know there are Democrats in the Senate. Or that Joe Biden (also a Dem) is publicly calling on Senate Dems to continue filibustering COVID relief because he thinks it will help them win Georgia,” replied Mr Cruz.
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