Theresa May is set to perform her second cabinet reshuffle in a week after the resignation of Priti Patel on Wednesday night and is likely to come under pressure to replace the Eurosceptic MP with another Brexit backer.

The former international development secretary resigned and acknowledged that her behaviour “fell below the high standards” expected of a minister after she attended meetings with senior Israeli figures while on a family holiday without reporting them to Downing Street.

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In her first public comment since her resignation, Patel tweeted a thank you message to her colleagues and constituents on Thursday morning.

Priti Patel (@patel4witham) An enormous thank you to friends, colleagues, constituents & the public for the support & kindness you have shown me over the last few days

The former Conservative leader Iain Duncan Smith, another Eurosceptic, said May should not seek to change the fine balance in the cabinet between remain and leave supporters.

“I don’t think Theresa May is looking to change the balance or send any great signal, that would be wrong,” he told BBC Radio 4’s Today programme. “It would be wrong, I think in her own mind, to make any great changes to the balance of the cabinet.”

Duncan Smith said May’s first priority would be “someone who is capable ... [who] you think will add value to the cabinet” but said their view on Europe would still be key.

“We are all Brexiteers now so the question is to what extent do you want someone in that job to be in support of David Davis and others? And I think on balance, having strong Brexit views, is one that I think the prime minister will certainly look for.”

Conservative MP Jacob Rees-Mogg suggested on Wednesday night that Patel’s downfall may have been linked to leaks from those within government who did not like her vocal support for Brexit.

“There are still some people who are still very bitter about the result a year ago and inevitably that colours their behaviour,” he told BBC’s Newsnight.

“If you go into how did Priti Patel’s visit come out in the first place, was it leaked by the Foreign Office, was it leaked by somebody at the Foreign Office who resented her and probably the foreign secretary’s role in Brexit, you may find something.”

Rees-Mogg said he did not view it as essential that Patel’s replacement was a long-time supporter of quitting the EU, but said it should not be “a frightful Eeyore”.

The prime minister may also seek to avoid the controversy that followed her appointment of Gavin Williamson as defence secretary following Michael Fallon’s resignation last week. Some of her MPs openly criticised her for promoting her chief whip and close ally, rather than widening the pool of talent around the cabinet table by promoting someone from the backbenches or junior ministries.

Patel was forced to quit her post after it emerged she had a series of 12 engagements with senior Israeli figures – including the prime minister, Benjamin Netanyahu – during a holiday in the country in August. It later emerged she had two additional meetings, one in the UK and one in the US, following her return from Israel.

The Israeli newspaper Haaretz also reported that during her stay in the country she visited an Israeli military field hospital in the occupied Golan Heights. Britain does not recognise Israeli control of the area seized from Syria.

Labour’s deputy leader, Tom Watson, wrote to May on Wednesday night demanding further details of Patel’s meetings in Israel, saying he had been told the former secretary of state met British consulate officials in Jerusalem, which he said made it plain that the Foreign Office did know she was in the country.

“I would like to know the facts of this case, because it is very unusual. I was told that the Foreign Office deliberately asked Downing Street to remove details of the briefing she received from Foreign Office officials when she was in Israel,” Watson told BBC Radio 4’s Today programme.

“If true, it shows that there was knowledge that Priti Patel was running a sort of independent foreign policy earlier, and that she’s not been sacked for breaching the ministerial code in doing that, but she’s been sacked because it became public that she was doing that.”