After patiently standing by through the first week of free agency, veteran tight end Jacob Tamme suddenly has two offers.

Tamme, who had 86 receptions and 14 special teams tackles in three previous seasons with the Broncos, is currently visiting with the Atlanta Falcons, according to an NFL source.

As Tamme was in Atlanta, the Broncos stepped in and made him an offer. Tamme had 52 catches for 555 yards in 2012, but his receiving role was reduced the past two years after much of his playing time went to slot receiver Wes Welker.

Tamme is seeking a greater offensive role. The Falcons’ primary tight end is Levine Toilolo, who had 31 catches last year in his second NFL season. With the Broncos, Tamme would settle in behind starter Owen Daniels and “blocking” tight end Virgil Green.

Mike Klis: mklis@denverpost.com or twitter.com/mikeklis