Cleveland needs a replacement for DE Genard Avery, who was traded to the Eagles on Monday for a 2021 fourth-round pick.
Moore, 27, is a former third-round pick of the Giants back in 2013. He lasted over two years in New York before he was waived in 2015.
The Dolphins wound up claiming him off of waivers from the Giants. Since then, Moore has played for the Raiders, Seahawks and Cowboys before returning to Oakland.
The Raiders cut him loose last year and he later signed on with the San Diego Fleet of the AAF. After the league folded, Moore signed on with the 49ers. San Francisco released him coming out of the preseason.
In 2018, Moore appeared in three games for the Cowboys and recorded five tackles and no sacks.
Horton, 29, signed on with the Panthers as an undrafted free agent out USC back in 2013. He was in the final year of his three-year, $1.485 million rookie contract when the Panthers cut him loose during the 2015 season before bringing him back soon after.
Horton returned to Carolina on a two-year contract before signing on with the Saints this past May. New Orleans released him back in September.
In 2018, Horton appeared in all 16 games for the Panthers and recorded 24 tackles and 1.5 sacks.