The Patriots shut them down when they made the same request a year ago - before the Texans hired Brian Gaine, only to fire him last week - and could play the same game this time around.
Schefter's tweet was short on details, but he noted that an National Football League investigation will follow.
The Texans would offer some familiarity for Caserio, with head coach Bill O'Brien and vice president of team development Jack Easterby both Patriots disciples.
According to ESPN's Mike Reiss and John McClain of the Houston Chronicle, Houston has officially requested to interview Caserio, but are awaiting a response from the Patriots. Revis later signed as a free agent with the Jets, who were only fined $100,000 for the tampering.
The NFL's definition of tampering is "any interference by a member club with the employer-employee relationship of another club or any attempt by a club to impermissibly induce a person to seek employment with that club or with the NFL".
Last Thursday night, the Patriots held their Super Bowl LIII ring ceremony at team owner Robert Kraft's home.
Do the Patriots believe that something happened between Caserio and Easterby at the party that led to Gaine's firing?