This may get confusing, so buckle up. The Falcons may well have asked Osi Umenyiora and his representation about taking a pay cut, but if ProFootballTalk is to be believed, they haven't boiled that down to take the cut or you're gone:
Per a source with knowledge of the situation, the Falcons routinely engage in an annual assessment of the roster, in light of player salaries and cap numbers. If/when that process flags a player who possibly is making too much money, the team inquires into the possibility of the player accepting a pay cut.
If the player isn't inclined to take less, then the Falcons will decide whether to keep the player or cut him; they won't squeeze the player to take less with an ultimatum.
It's a subtle but important distinction. The Falcons want players who buy in completely. They don't want players who will be disgruntled all season long after being backed into a corner.
So what does this mean? In essence, the report indicates that the Falcons didn't tell Osi he needed to take the cut or he'd be off the roster, which is the natural assumption for many of us when a story like this crops up. They asked Osi—and probably other players we haven't heard about just yet—to take those cuts, and they'll determine next steps based on the answers.
Practically speaking, this distinction only means that Osi isn't definitely getting his walking papers if he turns down a cut. At this stage of the game, with the Falcons clearly having at least brought it up to Umenyiora, you have to assume his release will be on the table if he says no. It's just slightly less of a black and white situation than it initially appeared.
You should consider these reports worth discussing but delicious with a grain of salt until we get some resolution on Osi's status, by the way.
How do you think this situation will resolve?