The deal includes $8.5 million in guaranteed money, the source said.
"NYC!!!!!" Taylor posted to Twitter, seemingly confirming the impending signing.
The Giants made obtaining a capable veteran backup QB a priority this offseason. They took a run at Mitchell Trubisky, who signed to likely be the starter in Pittsburgh, at the beginning of free agency.
It was imperative to the new Giants regime led by general manager Joe Schoen and coach Brian Daboll to avoid the type of offensive collapse the team experienced late last season when Daniel Jones was injured.
Taylor's role will be as a backup to Jones, whom the Giants have made clear they want to give at least one more chance as the starter. But it's not hard to envision Taylor getting on the field at some point this season. Jones, 24, has missed at least one game each of his first three professional seasons.
Taylor, who turned 32 last August, joined the Texans last offseason on a one-year deal worth up to $12.5 million and became the starter with Deshaun Watson not playing for the team after he requested a trade and 22 lawsuits were filed against him alleging sexual assault or inappropriate behavior during massage sessions.
After a strong Week 1 performance, Taylor injured his left hamstring in Week 2. He missed six games on injured reserve before returning in a Week 9 loss to the Miami Dolphins. Taylor struggled in his return, throwing five interceptions and taking 12 sacks in four games. He was benched in Week 13 against the Indianapolis Colts for rookie quarterback Davis Mills, who finished the season as the starter.
For the season, Taylor completed 60.7% of his passes for 966 yards, with 5 touchdowns and 5 interceptions.
Before joining Houston, Taylor signed a two-year deal with the Los Angeles Chargers in 2019 and, after playing behind Philip Rivers his first season there, was named the starter in 2020. That, however, lasted just one game because a team doctor accidentally punctured his lung with a pain-killing shot while attempting to treat a rib injury before the Week 2 game. When he was healthy enough to return, Taylor remained on the bench as the backup to rookie starter Justin Herbert.
In 78 career games, including 53 starts, Taylor has completed 61.3% of his passes for 10,736 yards, with 59 touchdowns and 25 interceptions.
ESPN's Jordan Raanan and Sarah Barshop contributed to this report.