Bellchambers, with forward Stewart Crameri, heads a list of Bombers who remain unsigned as this year's trade and free agency period grows closer.
"Obviously Tom Bellchambers is out of contract and I am extremely optimistic that we can get that one done," Dodoro told essendonfc.com.au.
"We are not far away at all in talking with his management."
The Bombers also confirmed AFL.com.au's report on Wednesday that it had upped its bid to keep the in-demand Crameri by offering him a longer deal.
The Western Bulldogs have tabled a four-year contract worth up to $1.8 million for the Bombers' leading goalkicker.
Crameri is likely to make a decision within the next few weeks.
"Stew Crameri has indicated all the way through that he would prefer to stay with us and we have offered him a three-year deal," Dodoro said.
"We have increased our offer but he does have a monster offer put in front of him from a rival club.
"We need to respect that and he needs to make a decision."
Essendon still has a number of other players out of contract, and Dodoro admitted some may seek more opportunities elsewhere.
Twitter: @AFL_CalTwomey