The actor, who plays the role of Edward Cullen's brother Emmett in the movie series, was reportedly offered the role of Spears' boyfriend in the video for her third single but declined, according to Hollywood Life.
The two stars recently met when Lutz was a guest at her fundraiser for the St Bernard Project's Hurricane Katrina relief effort.
The actor revealed to E! Online at the event that the track Oops!... I Did It Again is his favourite Britney song.
The video has recently been filmed in Los Angeles and would have seen Lutz rescuing Spears from the paparazzi before kissing in a car.