Following his conviction at Southwark Crown Court Khan, 48, said he would resign from the seat he has held since 2019.
It is understood this process has not yet begun.
Khan is still awaiting sentence and can still serve as an MP, despite his conviction.
But now Ossett Tory councillor Tony Homewood, who stood as the MP's election agent, has asked Khan's family to intervene to get him to fulfil his promise to resign.
Writing on twitter Coun Homewood said: "It is with considerable regret that I have noticed that @imranahmadkhan has not yet applied for the position of Crown Steward & Bailiff of the Chiltern Hundreds (ie; resigned as an MP) despite giving a public undertaking to do so.
"I know Mr Khan's family, having met them all. His brothers, Khalid and @KarimKhanQC, are both men of the utmost honour and integrity in my experience.
"Why their brother continues to drag the otherwise good name of their family through the mud shows he has little more respect for them and their mother, than he does for the electorate of #Wakefield and their late father, a respected and fondly remembered local doctor. I realise that @KarimKhanQC Is involved in very important work at the ICC but I implore him to intervene in this matter.
"Karim...your brother needs to keep his word for once and resign. I apologise for having to involve you in this way but you and Khalid are the only people who have the slightest influence on him. This is about him claiming money from the state for as long as possible.
"I would remind everyone reading this tweet... that the Khan family are good Wakefield people and are not responsible for the appalling behaviour of one of their number, which I can assure you they will certainly not condone. We pick are friends but we can't pick our relatives."
Imran Ahmad Khan was expelled from the Conservative party after his conviction.
To trigger his resignation Khan needs to write to The Treasury, and then a date for a by-election can be set.
Imran Ahmad Khan became Wakefield MP following the 2019 election, and was the first Tory in more than 89 years to take the so-called Labour safe seat.
Khan, who was born in Wakefield, stopped Labour’s Mary Creagh from being re-elected.
He won 21,283 votes and ended her 14 years as the city’s MP. Ms Creagh polled 17,925 votes.
Yorkshire Party candidate Ryan Kett won 868 votes and Liberal Democrat candidate Jamie Needle won 1,772.
Independent Stephen Whyte won 454 and the Brexit Party’s Peter Wiltshire won 2,725.
Labour had held the constituency since 1932.