Sure you can. We franchised and traded John Abraham in 2006.
Remember last year the Eagles tagged DJax and tried to trade him for anything they could get, they didnt get ny offers, removed the tag, and released him. Reportedly, they were asking for a 3rd rd pick for the franchised WR. here's the article:
http://www.nfl.com/news/story/09000d5d827568b5/printable/franchise-season-making-sense-of-tags-and-tenders
"The franchise tag allows teams to protect their investment -- which is a significant value. Even if a team does not particularly want to keep a player, or feels a long-term deal is hopeless, it can tag the player and put him on the trading block, knowing the value of the player is greater than the compensatory third-round pick the team could receive for losing a high-caliber player to free agency.
For example, let's examine the Eagles' situation. Philadelphia has enough salary-cap room to tag Jackson, but that doesn't necessarily mean he'll be donning midnight green in 2012. At the NFL Scouting Combine, Eagles brass broadcasted a clear message around the league that they would listen to trade offers for Jackson. This wasn't something that slipped out from a rogue source; Philadelphia basically took out a full-page ad in the paper announcing Jackson's availability. Teams don't make intentions this public if they are not truly interested in moving a player.
Now that Jackson's franchised, the Eagles are hoping a team will offer them more than a third-rounder -- the probable compensation if Jackson were to leave in free agency and Philly didn't go on another free-agency binge. (Compensatory picks are awarded to teams that lost more free agents than they gained.) Philadelphia initially will claim it wants a first-rounder, but privately will take a little less. Long story short: By franchising Jackson, the Eagles put themselves in position to possibly gain a better draft pick if the speedy receiver departs.
However, the Eagles will play hard ball with any potential Jackson trade. With cap room available, they can bring Jackson back for the franchise number if no enticing trade options materialize. This would force the mercurial receiver to prove his worth in a potential walk year. And if he does, Jackson could earn the long-term contract he desires. By tagging Jackson, the Eagles are in full control and don't have to make a deal they don't want to make."