Chris Tappin recently failed in his final attempt to avoid extradition to the US at the European Court of Human Rights.
The retired British businessman from Kent is accused of conspiring to sell components for Iranian missiles, but maintains that he did not know the batteries shipped through his freight company were destined for Iran.
Talking to Total Politics, Jo Johnson described the case as "a constitutional issue with national twist".
He suggested that his constituent would be under "incredible pressure" to agree to a plea bargain to "avoid lengthy incarceration" before a trial.
Johnson added that Tappin could be seen as an example of "British middle-class respectability" who was to be "catapulted robotically to the US".
Tappin is expected to testify before the home affairs select committee about the US-UK extradition treaty, and the evidence session may actually delay his enforced departure from the UK.
Tappin is not alone in fighting extradition to the US. Other cases including 23-year-old student Richard O'Dwyer, who is accused of breaking US copyright laws, and computer hacker Gary McKinnon.
Sir Scott Baker's independent review of the UK's extradition arrangements last year concluded that the current treaty between the US and the UK was both balanced and fair.
But critics believe this is the latest case to test the review's findings.
Johnson will raise the case at PMQs later today.