Two Tory deputy chairmen have stepped down from their party positions after backing amendments that seek to beef up Rishi Sunak’s Rwanda Bill.

Lee Anderson and Brendan Clarke-Smith said that “whilst our main wish is to strengthen the legislation, this means that in order to vote for amendments we will therefore need to offer you our resignations from our roles”.

In a joint letter, they told the UK prime minister it was “important in terms of credibility that we are consistent” on arguing that safeguards must be put in place to ensure the British government’s flagship asylum policy is legally watertight.

It comes as Rishi Sunak was dealt a fresh blow to his authority as 68 MPs, many likely to be from within his own party, voted in favour of changes to the Bill put forward by Conservative backbencher Bill Cash.

The amendment seeking to ensure UK and international law cannot be used to block a person being removed to Rwanda was rejected by a majority of 461, but the rebellion gives an indication of the scale of unease within the Conservative Party during an election year.

Around 40 Tory backbenchers including former prime minister Liz Truss, former ministers Suella Braverman, Jacob Rees-Mogg and Simon Clarke and former leader Iain Duncan Smith earlier met to agree backing the changes.

Jane Stevenson, a parliamentary private secretary in the British Department for Business and Trade, also confirmed she would vote for the amendment.