Cary Grant was the first man who had a chance to get the James Bond role. He was the first choice of producers when casting was being done for the first official Bond movie, Dr. No. However, he said that he would only do one movie, and not a series. Because of this, he was not given the role.
James Mason was the next choice after Cary Grant. However, he said he would only do two movies, and no more. Since this was still not satisfactory to the producers, other actors were looked to.
John Frankenheimer, now known as the director of Reindeer Games and Ronin, says he was offered the role of James Bond in 1962 by Broccoli himself. However, Frankenheimer did not want to be in the larger-than-life type of movie that the Bonds were to be.
Adam West was offered the Bond role by Cubby Broccoli. This was before On Her Majesty's Secret Service came out. West seriously considered taking the part, but ultimately decided that Bond should be played by an Englishman.
Sir Michael Gambon, who was a very respected star of stage back in the 1960's and 1970's, was offered the role of Bond. He looked nothing like Bond, but Broccoli felt this didn't matter. However, Gambon never got the part.
Patrick Mower was in consideration for the part at the same time Gambon was. Mower was a relatively unknown British actor. The producers decided that they did not want to cast an unknown, so Mower never got the part, and John Gavin was looked to.
John Gavin was very close to becoming Bond. He was actually given a holding contract, and would have been Bond except for the return of Connery. Gavin was paid fifty thousand dollars for the holding contract. In a way, he was paid for the role of Bond.
After Connery made it clear that he was not going to return after Diamonds Are Forever, Burt Reynolds was mentioned in connection with the Bond role. He had heavy support from MGM. However, many people would have considered his casting a marketing ploy, and there was still the desire to have Bond played by an Englishman.
With the possible exception of John Gavin, James Brolin perhaps became the closest to ever becoming Bond without actually landing the role. Brolin was already a star because of Pensacola: Wings of Gold. He was screen tested for Bond in Octopussy, since Moore was holding out his contract for more money. Brolin would almost certainly have become James Bond had not Moore finally come to terms and returned.
Sam Neill was later screen tested for the role of James Bond. He seemed to have a good chance, but sadly this never panned out. In my personal opinion, Neill would have been the best Bond out of all of the "Almost Bonds".
In the early 1980's, Lewis Collins had huge fan support to become James Bond. However, this support did not extend to the decision makers. The producers of the movies did not like Collins, and in the end it was their decision that he not get the part.
In the 1980's, Rutger Hauer was asked several times to play a Bond villain. Whether he was up for the role of Bond at any point is questionable, but a distinct possibility.
Michael Billington was once in the running to be James Bond. While he never made it, he did make it into a Bond film, appearing as Anya's lover in The Spy Who Loved Me.
Alan Rickman tested for the role of James Bond in 1993. When that didn't work out, he was supposedly offered the role of Alec Trevelyan. However, he turned that role down.
Sean Bean, now known as Alec Trevelyan to Bond fans, actually tested for the role of James Bond before being cast as his nemesis.
Colin Wells was seen by someone from EON at Glasgow Citizen's Theater. His acting and his looks were apparently good enough to be given a screen test in 1994.
Mel Gibson was reportedly offered the role of James Bond in 1992. He claims to have turned the role down because it was too frivolous. However, EON denies that he was ever offered the role.
When the 1967 Casino Royale spoof was being cast, some of the names to be considered were Stanley Baker, William Holden, Peter O'Toole, and Laurence Harvey.
Throughout the years, Kevin McClory has been trying to get another remake of Thunderball done. Known at one time or another as SPECTRE and Warhead, the movie would have to feature a new James Bond. At one point or another the following names were connected to the role of Bond for the movie: Lewis Collins, and Sam Neill (both of whom were once in contention for the role in an official movie), Pierce Brosnan (who later became Bond), Christian Burgess, Liam Neeson, George Clooney, and Sean Connery's son, Jason Connery.
Lastly, there have been many other people who have been mentioned in some way as having at least a remote chance of becoming Bond. Some of these men had a better shot than others. Many however, were just mentioned as a possible choice but never pursued:
William Franklyn, Patrick McGoohan, Michael Craig, Peter Snow, Robert Campbell, Anthony Rogers, Hans de Vries, John Richardson, Paul Newman, Steve McQueen, Michael Caine, Julian Glover (who became Kristatos in For Your Eyes Only, Jeremy Brett, Tom Selleck, Greg Wise, Dave Warbeck, Anthony Andrews, David Robb, Michael Jayston, Ian Oglivy, Nicholas Clay, Oliver Tobias, Gordon Williams, Michael Petrovich, Gary Myers, John Ronane, Michael McStay, Simon Oats, Roy Thinnes, Patrick Allen, Richard Burton, Richard Johnson, Lambert Wilson, Finlay Light, Andrew Clarke, Simon McCorkindale, John James, Michael Praed, Mark Greenstreet, Niel Dickson, Bryan Brown, Steve Adler, Ben Cross, Charles Dance, Michael Nader, Marcus Gilbert, Clive Owen, Ioan Grofudd, Jonathan Cake, Paul McGann, Linus Roache, James Purefoy, Nathaniel Parker, Ralph Fiennes, Jason Isaacs, Jeremy Northam, Mark Frankel, Adrian Paul, and Hugh Grant.