The England international has been a target for the Red Devils for a number of seasons, most recently having an enquiry rejected by Diego Simeone's side during the last campaign.
Aaron Wan-Bissaka's form for Man United has been questioned, particularly after he gave away the penalty which Chelsea equalised from on Sunday, and Trippier would be a direct replacement in the full-back role.
Manchester Evening News reports that Rangnick will decide whether to pursue a deal for Trippier once he has fully started his role with the Red Devils, as he is currently waiting for his work permit.
The 31-year-old has provided 11 assists in 83 appearances for Atletico Madrid, and won La Liga last season, his first major trophy in his career. body check tags ::