The 27-year-old had previously been in talks over a potential switch to Scotstoun with head coach Franco Smith known to be an admirer of Kriel, having worked in South Africa earlier in his career.
However, Forrest has confirmed the move will not go ahead with reports in South Africa indicating Kriel is now likely to sign for La Rochelle.
"Yes, that one's dead," Forrest told The Scotsman ahead of Saturday's Champions Cup meeting with Bulls.
"I think there were some conversations going on. Probably there's been quite a few conversations, as there always is when you sound out players.
"Everyone knows we've got guys moving on at the end of the season, so we're always looking at where do we need to strengthen, what young players have we got coming through, which obviously we've seen quite a lot in the last couple of weeks, like Fergus Watson and Johnny Ventisei stepping up for their first cap."
Despite expressing satisfaction with the contributions of emerging players, Forrest indicated that the club is still seeking experienced additions.
Glasgow are preparing for a significant squad reshuffle, with a group of internationals leaving at the end of the campaign.
Huw Jones will join Toulon, Jack Dempsey is set for a move to Japan, Adam Hastings has signed for Montpellier and Jamie Bhatti has agreed a return to Bath Rugby.
Forrest added: "We know we've got young players coming through, but we're also looking to see where we can strengthen as well."