The Scotland international's current deal expires in 2022, but he is in advanced talks to further extend his stay.
Forrest has gone from strength to strength along with the team in recent seasons and, after nine consecutive domestic trophies, he feels they will only get better with the improved challenge from Rangers.
Manager Neil Lennon had earmarked a new deal for Forrest at the start of the season and the winger said: "It's near enough done, everything is basically agreed. I'll be delighted to get that done really soon and just keep focusing on playing.
"I've always stated from the start of pre-season that I want to stay at Celtic and the club have done the same thing. The talks have been really positive and it's maybe near the end."
Lennon handed Forrest his debut in May 2010 and made a new contract for the wide player one of his priorities after finalising his return to the club on a long-term basis.
"It gives me confidence when I'm playing," Forrest said. "It's a great team to be involved in just now and I just want to keep working hard to stay there."
The 28-year-old took his tally for the season to nine goals with a strike in Celtic's 4-0 win over Aberdeen on Sunday.
The Hoops had dropped five points in their previous two away games in the Ladbrokes Premiership but have bounced back in style by netting 10 goals in two league games either side of their Europa League win over Lazio.
"We had a great result against Cluj and a good performance and then a couple of days later was the Livingston result," said Forrest, who was promoting a sponsorship deal by BeCordial Hotels with Celtic Women.
"After the game the other night we were all wanting to make sure the league is the priority and we got back on it with another good performance and result.
"The manager said after Thursday was such a great night for the players, the staff and the fans, a couple of days later you have another big game at Aberdeen in the league and if you don't win you get questioned.
"So it makes Thursday night even better when you go to Aberdeen and win as well a couple of days later."
Celtic regained top spot from Rangers the previous weekend by thrashing Ross County but they are only ahead on goal difference.
"When you look at the last couple of years the league has been tighter and other teams have strengthened as well," Forrest said. "It's really close just now and both teams are doing well in Europe.
"It is competitive, we know that, but we just need to stick to our own game and worry about ourselves. We are playing really well and just need to keep that form up.
"It's good as well because you raise your standards and raise your game as well when you know you can't afford to slip up, when you know other teams round about you will punish you.
"It keeps the edge and it makes us go into every game knowing that we need to be at the top of our game to get the three points.
"You have a big game every three days so you can't afford to rest on what you have done the game before. You need to push on and the coaches are good at getting that out of the players."