The 33-year-old Greening, who has played in the Premier League for Middlesbrough and West Brom, will stay at Oakwell until January 2.
Boss Keith Hill hopes the former Manchester United player will have a calming influence in the Reds' midfield.
"I'm looking for Jonathan to be our handbrake on the enthusiasm we have, someone to control things, to instil an intelligent energy and bring everything together," Hill told the club's official website.
"He has an abundance of ability, a mentality that is Premier League, it's not something that you lose. He will be able to harness and embrace the energy we have as a team on pitch."
The 30-year-old Buzsaky, who has 20 caps for Hungary, began his career with MTK Hungaria and arrived in England from Porto when he joined Plymouth in 2005.
He signed for QPR for £500,000 in January 2008 following an initial loan spell at Loftus Road and moved on to Portsmouth - where he has been a regular this season - in October last year.
He becomes Keith Hill's fourth signing of the week after the captures of Greening, Marcus Tudgay, also from Forest, and Peterborough's Emile Sinclair.