However, the NSW team's chances of doing so have been hit by the enforcer's suspension, imposed for a dangerous tackle on Jake Strachan 21 seconds into Saturday's victory over Western Force.
Strachan was ruled out of the second half with concussion.
Waratahs are currently ninth, but on equal points with Fijian club Drua in the eighth-and-last finals qualifying spot.
The ban is also a blow to the 26-year-old's World Cup aspirations. The flanker, who last played for the Wallabies in 2021, was called into the international squad on Sunday morning.
Swinton's tackle was not sanctioned at the time but a SANZAAR Judicial Committee hearing imposed the ban after meeting on Tuesday.
"Having conducted a detailed review of all the available evidence, including all camera angles and additional evidence, including submissions from the Player and his legal representative, Mr Aaron Lloyd, the Judicial Committee upheld the Citing under Law 9.13," committee chair Helen Morgan ruled.
"With respect to sanction, the Judicial Committee deemed the act of foul play was reckless, with the contact point directly made with the head, high force and no mitigating factors present.
"The Committee found the incident was highly dangerous and, after considering all relevant factors of World Rugby's Head Contact Process and sanctioning table, decided the foul play merited a top-end entry point of 10 weeks.
"Taking into account mitigating factors, including the Player's acceptance of guilt that the incident was foul play, conduct at the hearing and remorse, the Judicial Committee reduced the suspension by three weeks.
"The player is therefore suspended for 7 weeks, up to and including 3 June 2023."
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