The prosecutors, Salt Lake County District Attorney Sim Gill, a Democrat, and Davis County Attorney Troy Rawlings, a Republican, said federal investigators should look into the alleged impropriety, which goes beyond their ability to investigate or prosecute.
Reid's office said Monday that neither the senator nor his aides have spoken to investigators.
Lee's name appeared in the final report from the Utah House Special Investigative Committee released last week.
The report touches on Swallow's efforts to help raise money for Lee's successful 2010 campaign, including rounding up donations from St. George businessman Jeremy Johnson and a number of Johnson's associates.
Johnson now faces scores of federal felony fraud charges stemming from his I Works business, which prosecutors allege scammed customers out of $275 million.
"John Swallow, as you know, has been involved in Utah politics for a long time," Lee said. "Shortly after I announced my candidacy for the Senate in 2010, John Swallow was one of the many people who approached me and said, 'I'd like to be helpful,' and he was in [fundraising] and other ways.
"He himself was supportive," Lee added, "and introduced me to other people who were supportive."
In a February 2010 email from Swallow to Shurtleff, Swallow asks his then-boss: "can i [sic] introduce Mike Lee to Jeremy Johnson?" Shurtleff said he could.
By June, the report says, Swallow was "working hard" to raise money for Lee. In an email the day before the 2010 primary election, Swallow notified Johnson that four checks from Johnson associates had bounced. Johnson apologized and said, "I will get it fixed ASAP. Let me know whos [sic] bounced. I was in a mad rush to get those in so maybe I pushed a few people too hard."
Thomas Burr contributed to this report.