Bellshill MP Phil Boswell is facing a call to quit just eight months after he was elected as a parliamentary investigation into his conduct begins.
The SNP politician is accused of breaking strict rules by failing to declare his role as a company director.
Mr Boswell is under pressure to step down from the SNP group at Westminster while the probe is carried out.
However, North Lanarkshire Council leader Jim McCabe believes he should consider leaving the House of Commons altogether because of his “hypocrisy”.
Mr Boswell took the Coatbridge, Chryston and Bellshill seat from veteran Labour MP Tom Clarke in May.
Last month he was under fire after it was revealed he had used a tax avoidance scheme to receive a loan of more than £18,000 while working as a manager in the energy industry.
Earlier in the year the MP had been pressing the Treasury to close loopholes to prevent tax avoidance.
There was no suggestion he had acted illegally, but now the Parliamentary Commissioner for Standards has confirmed an investigation will take place after a complaint by Paul McGarry, a Lib Dem candidate for this year’s Scottish Parliament elections.
This concerns a claim that Mr Boswell failed to record in the parliamentary register that he was a director of a company, Boswell and Johnston Limited.
Mr Boswell could not be contacted for comment.
Since May two newly-elected MPs, Michelle Thomson (Edinburgh West) and Natalie McGarry (Glasgow East), have withdrawn from the SNP whip while separate probes into alleged cash irregularities take place.
Councillor McCabe stressed he did not wish to “judge” Mr Boswell, but added: “This situation stinks of hypocrisy and he should consider his position as an MP.
“He said he was strongly opposed to tax avoidance, but was then found to be participating in it himself.
“He seems to have used his position. You can’t tell people to do one thing and do another yourself.”