UPDATE: A signing isn't expected, at least not yet:
LB Jonathan Vilma won't be signing with the Falcons after today's workout. If it happens, it'll be down the line (e.g. training camp).— Mike Garafolo (@MikeGarafolo) June 18, 2014
This isn't surprising. The Falcons have a dearth of veteran options at linebacker at the moment, and that's especially true at inside linebacker, where Akeem Dent is the improbable elder statesman. Vilma may not be an ideal fit inside in 3-4 sets, but he can move around a little in the formation and has made a career, as Murf Baldwin has noted, as a linebacker who excels in pursuit and doesn't leave you high and dry in coverage. If those skills are still there, he's an asset, even if he's likely to just be a reserve at age 32.
The question is health and whether his injuries have further sapped his effectiveness, with Saints fans telling you he's pretty well washed up. If the Falcons do sign him, we'll have to hope he's able to contribute in a meaningful way and provide veteran savvy and talent to a group of linebackers that has real talent but plenty of question marks.
Worth noting that the always reliable Vaughn McClure doesn't think Vilma is inevitable:
I wouldnt count on Vilma signing.— vaughn mcclure (@vxmcclure23) June 18, 2014