England, who are leading the series 2-0, will go into the third Test, starting on August 1, with a 14-member squad.
Finn had played the first Test and was replaced for the second by Tim Bresnan. Onions did not get a game.
Also back in the squad are batsman James Taylor and paceman Chris Tremlett. Taylor is in the squad as a cover for Kevin Pietersen, who had sustained an injury during the second Test at Lord's.
National Selector Geoff Miller said on Sunday (July 28): "Kevin Pietersen is continuing his recovery from a calf strain and he will be assessed by the medical team closer to the start of the Test and we have therefore included an extra batsman in James Taylor who has been in good form for Nottinghamshire this season.
"Chris Tremlett has worked hard to regain fitness and form following a couple of injuries and his performances for Surrey this season have been very encouraging.
"Including an additional spinner in Monty Panesar who has plenty of international experience provides Alastair Cook and Andy Flower with a number of options."
Alastair Cook (captain), Joe Root, Jonathan Trott, Kevin Pietersen, Ian Bell, Jonny Bairstow, James Taylor, Matt Prior (wicketkeeper), Tim Bresnan, Stuart Broad, Graeme Swann, James Anderson, Chris Tremlett, Monty Panesar.