The German has caught the eye of a number of clubs domestically and abroad following his work with the Terriers this season, taking them from outsiders in the Championship promotion race at the start of the year to a place in Monday's playoff final.
According to the Sunday Mirror, Garry Monk, who resigned from Leeds United three days ago, remains Boro's top managerial target but the club will turn their attentions to Wagner should Monk turn them down.
The newspaper claims that Boro owner Steve Gibson will target Wagner regardless of whether the Terriers earn promotion to the Premier League as he eyes an immediate return to the top flight.
Gibson is thought to have promised "extensive funds" to the cause and will wait to see the outcome of the playoff final, in which Huddersfield face Jaap Stam's Reading, before taking the next steps in appointing a new manager.
Boro finished last season 19th in the Premier League, 12 points adrift of safety.