In a letter to the chair of the Treasury select committee Mel Stride, the 31 MPs say the Chancellor's contact with financial traders in the run up to the policy change should be scrutinised "without delay".
Mr Kwarteng was reported to have privately met with at least one trader who profited from the fall in the value of the Pound in the weeks ahead of the Budget.
He has also admitted attending a private party of financiers just hours before delivering his speech to the Commons, where he was supposedly "egged on" by hedge fund managers.
The MPs said that any "suggestion or suspicion of private individuals using privileged access to profiteer from our country's economic difficulties must be fully investigated".
Signatories of the letter include Dame Margaret Hodge, who is a former chair of the Public Accounts Committee, and John McDonnell, the ex-shadow chancellor.
"As Members of Parliament and public servants to our constituents, we are writing to you to request that, as Chair of the Treasury Select Committee, you use all powers at disposal to investigate allegations of insider trading preceding the recent collapse of the value of the pound against the US Dollar," the 31 Labour backbenchers said.
They added: "Any suggestion or suspicion of private individuals using privileged access to profiteer from our country's economic difficulties must be fully investigated. Our constituents are already deeply angry that financial traders are enriching their personal wealth by betting against Britain. Any political assistance received in that process would be unacceptable.
"We therefore urge you to begin, without delay, an inquiry of the Treasury Select Committee into any and all contact between the Chancellor of the Exchequer, Prime Minister, and financial traders in recent weeks. Transparency is sorely needed."
Mr Kwarteng's budget included tens of billions of pounds in unfunded tax cuts, which caused an immediate tanking in the value of the pound, and a surge in the yields on government debt.
The unusually sharp market movements left ample scope for traders to profit by shorting Sterling, but also caused significant financial instability – undermining the asset bases of pension funds.
Interest rates are expected to surge to as much as 6 per cent as a result of the policies, despite a partial U-turn by Mr Kwarteng and Liz Truss on the abolition of the top rate of tax.
Prior to entering politics Mr Kwarteng worked as a City financial analyst, as well as for a hedge fund.
Labour MP Karl Turner, one of the letter's signatories, told The Independent that the allegations regarding the budget were "incredibly serious".
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"If these allegations were made in any organisation outside of Government is would very probably be a matter that the police were asked to investigate," he said.
"However, these allegations are even more serious. The Governments mini budget me at that the markets reacted requiring the Bank of England to take immediate action.
"To think that the chancellor’s cronies and Tory donors were betting against Britain with privileged inside information is utterly scandalous. We must get to the bottom of these allegations quick sharp.”
Separately, Jo Maugham, the director of the Good Law Project, said his organisation would "take legal advice on what needs to be shown to establish criminal insider dealing by the Chancellor" before considering next steps.
He added that the group, which has a track record of launching judicial reviews into contentious government policies and decisions, was "not afraid to push regulators".
The Independent has contacted the Treasury for comment on this story. The Financial Conduct Authority declined to comment on the claims.