According to Cincinnati.com, the Bills will receive the Bengals' first-round pick, which is No. 12 overall, while the Bengals will get either the No. 21 or No. 22 overall pick, both of which are held by the Bills.
Trade agreements cannot be made official until Wednesday.
Glenn, 28, played just six games due to foot and ankle injuries last season, but he has started 77 of the 78 games he has played since 2012.
Glenn signed a five-year, $60 million contract with the Bills in May 2016, but that was before Brandon Beane became the Bills general manager and Sean McDermott became their head coach.
In recent weeks, Bengals owner and president Mike Brown stressed the need to address the offensive line.