One of the men claimed he was raped by the Rev. James Hanlon in 1988 and 1989 and believes Hanlon told Teczar about his vulnerability. The unnamed plaintiffs were ages 12 to 18 at the time.
Diocese spokesman Pat Svacina told The Associated Press on Wednesday that the settlement, reached through mediation, involves only the claims involving Teczar.
The diocese, citing a request by the victims, said it would not disclose details or the amount of the settlements.
Bishop Kevin Vann said he personally, and in the name of the church, apologizes for abuse encountered at the hands of Teczar.
"The sexual abuse of anyone, especially minors, is repulsive to me," Vann said in a written statement. "It is a sin and a crime. That it is done by a priest is disheartening."
After a March trial in Eastland, Teczar was sentenced to 25 years in prison for raping and molesting an 11-year-old boy in the early 1990s.
In 2005, the Fort Worth diocese settled for $4.15 million a lawsuit with two of Teczar's accusers, including one who filed the criminal complaint in Eastland.
Hanlon died of cirrhosis of the liver in 1990, records show. Abuse allegations against him and five other priests are detailed in 700 pages of previously secret diocese documents released by a judge last year after the lawsuit was settled.