Stephen Timms, chairman of the Work and Pensions Committee, had previously raised concerns that the bill did not address the problem of online scams and said he was willing to step in to force change.
Advisers have also started writing to their MPs, asking them to urge the government to include financial harms in the bill.
Meanwhile Conservative MP Ruth Edwards said there was a need for internet companies, including search engines and social media sites, to patch loopholes in their verification procedures and prevent scammers from using their platforms for illicit activity.
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