Siddiq, whose ministerial role included tackling corruption in UK financial markets, was named last month in an investigation into claims her family embezzled up to £3.9bn from infrastructure spending in Bangladesh.

Her aunt is the former Bangladeshi prime minister Sheikh Hasina, head of the Awami League, who fled into exile after being deposed last year.

Siddiq has also come under intense scrutiny over her use of properties in London linked to her aunt’s allies.

The MP insists she has done nothing wrong, but the prime minister has faced calls from Conservative leader Kemi Badenoch to sack her as a minister.

In a letter accepting her resignation, Sir Keir said the “door remains open” for her.

Sir Laurie Magnus, the independent adviser on ministerial standards, said he had “not identified evidence of improprieties” connected with Siddiq.

In a letter to the prime minister, Sir Laurie said: “A lack of records and lapse of time has meant that, unfortunately, I have not been able to obtain comprehensive comfort in relation to all the UK property-related matters referred to in the media.

“However, I have not identified evidence of improprieties connected with the actions taken by Ms Siddiq and/or her husband in relation to their ownership or occupation of the London properties that have been the subject of press attention.

“Similarly, I have found no suggestion of any unusual financial arrangements relating to Ms Siddiq’s ownership or occupation of the properties in question involving the Awami League (or its affiliated organisations) or the state of Bangladesh.

“In addition, I have found no evidence to suggest that Ms Siddiq’s and/or her husband’s financial assets, as disclosed to me, derive from anything other than legitimate means.”

But he said it was “regrettable” that Siddiq “was not more alert to the potential reputational risks” of her close family’s association with Bangladesh.