Tavernier, a 23-year-old right-back, and 25-year-old striker Waghorn, appear set to follow Andy Halliday in joining the club after the former Middlesbrough and Bradford winger signed a two-year contract on Saturday.
Former Latics team-mate Rob Kiernan signed on at Ibrox earlier in the summer along with ex-Liverpool and Hearts defender Danny Wilson.
Waghorn had a decent spell at Leicester City under former England manager Sven Goran Eriksson and was selected for England U21s before going on loan to Hull, Millwall and then Wigan, who he joined permanently in April 2014.
Tavernier came through Leeds United’s youth system before joining Newcastle United in 2008, making his first team debut a year later.
The pair are the latest young players brought in from England by new Rangers manager Mark Warburton, who took Brentford from League One into the English Championship where they last season enjoyed their highest second-tier finish for 80 years.