Solder has started every game over the past two seasons with the Giants since signing a four-year, $62 million deal as a free agent.
While the Giants (2-5) have been only slightly worse than the Browns (2-4) this season, Cleveland could still be more aggressive on the trade market after entering the year with high expectations.
As La Canfora noted, the two teams have already have a good relationship after making several moves last offseason.
In any case, the Browns certainly need an upgrade at left tackle after a disappointing effort from Greg Robinson to start the year. The veteran is in "serious jeopardy" of being benched when the team returns from its bye this week, per Mary Kay Cabot of Cleveland.com.
Cleveland has allowed 16 sacks as a team in six games, and it has contributed to Baker Mayfield's struggles to start the year.
Cabot also reported the team's interest in Williams, but the Washington Redskins appear unwilling to make a move even though the Pro Bowler has been a holdout since training camp.