NFL Network's Michael Lombardi reports that Shane Andrus, Steve Hauschka, Connor Hughes and Parker Douglass also tried out for the job.
Folk leads the NFL with 10 missed field-goal attempts, going 18 of 28 and missing seven of his past 11.
Suisham is 85 of 107 (79.4 percent) on field-goal attempts in his NFL career. He originally signed with the Pittsburgh Steelers out of Bowling Green in 2005, went to the Cowboys' practice squad and was signed to the active roster on Oct. 24. Suisham played in three games and was 3 of 4 before being released.
Back with Dallas in 2006, Suisham played in three games and was 1 of 2 before again being cut. He finished the season with the Redskins, appearing in five games and converting 8 of 9 field-goal tries. Suisham played in 16 games for the Redskins in 2007 (29 of 35) and 2008 (26 of 36) and 12 games in 2009.
Dallas tried several remedies while Folk piled up a six-game streak of at least one miss.
First, the Cowboys switched holders, replacing Mat McBriar with Tony Romo even though their quarterback is haunted by a critical flub late in a playoff loss at Seattle three years ago. Then they tried an in-house competition with kickoff specialist David Buehler, who proved unreliable on field goals.
Concerns eased when Folk nailed a 44-yarder just before halftime against the Saints, but it didn't take them long to come back.
"It's a mystery to us," Phillips said before Folk was cut. "We have tried to work it out a lot of different ways. Now it's come to this point."
Adding to the mystery, Folk had missed just 10 kicks in his 2½-year career before slumping. Television footage during Saturday night's game showed Folk's pregame kicks flying all over the place in the controlled atmosphere of the Superdome.
The Associated Press contributed to this report.