Clowney is not the only target in mind to improve the defense, as Paul Schwartz of the New York Post reported interest in linebacker Jordan Jenkins, who played for the New York Jets the last two seasons.
Defense had to be the top issue to address in the offseason since the Giants were one of nine NFL teams to concede over 6,000 total yards last season.
New York also allowed the fifth-most passing yards and 13th-most rushing yards during a disastrous 4-12 campaign that was the last for Pat Shurmur as head coach.
The Joe Judge era needed to start with a focus on improving a unit that failed to provide much support to Daniel Jones, Saquon Barkley and the offense.
The Giants took the first step in achieving that Monday by agreeing to a three-year deal with defensive back James Bradberry, per ESPN's Adam Schefter.
Because the NFC East side has an abundance of salary-cap space to work with, the $45 million deal, with $32 million guaranteed, for Bradberry will not hamper their pursuit of other marquee free agents.
The team's NFL draft position could also have an effect on what the direction is in free agency.
Barring an unforeseen turn of events, the Giants will not be able to take Ohio State's Chase Young with the No. 4 overall pick.
Instead of bringing in a young, cheap pass-rush improvement, the Giants may have to spend more to fix that issue.
Teaming Clowney with 2019 first-round pick Dexter Lawrence and Dalvin Tomlinson would solidify the interior and allow the Giants to focus on shoring up the pocket for Jones, or on Clemson's Isaiah Simmons in the draft.
As long as Clowney remains healthy, he would be a massive upgrade for the Giants, and he could help them make a move up the standings.
But the Giants should face competition for his signature, as one of Seattle's offseason goals was to bring him back, as ESPN's Jeremy Fowler reported.
The pursuit of Jenkins would likely cost less, even though he is coming off a pair of strong seasons with the Jets.
If Jenkins inks a deal with the Giants, he would join David Mayo in a linebacker group that had four of the team's top six sack-producers last season.
A free-agent move for Jenkins likely would not hamper the pursuit of Simmons in the draft because he might be the most versatile defensive prospect available.
Simmons played defensive end, linebacker and safety during parts of Clemson's season, so he could be used in all facets of the game plan, if the Giants go in that direction April 23.
With steps already made to improve the defense, and a few potentially coming soon, the Giants could look much more competent on that side of the ball in Judge's first year in charge.
Follow Joe on Twitter, @JTansey90.
Statistics obtained from Pro Football Reference.
Contract information obtained from Spotrac.