Midfielder Brown, 33 signed from Hibernian in 2007, while striker Griffiths, 28, joined from Wolves in 2014.
Winger Forrest, 27, came through the Hoops’ youth academy and made his first-team debut in 2010.
Ahead of the trip to St Mirren in the Ladbrokes Premiership on Friday night, Rodgers said: “We are pretty close on Leigh. He clearly wants to be here, we want him to be here, so that will be good news.
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“We think that it is not too far away.
“Scott has made it clear he wants to be here and I am sure there will be a deal done pretty soon on that. “He is quite relaxed about it. He knows what we think of him here at the club so hopefully that will be done.
“James and his people will be speaking with the club and, likewise, a player we would love to extend here because he has been absolutely brilliant in my time here.”