Since he is on their roster, there is nothing stopping them holding talks with his agent regarding his willingness to sign. There is also nothing stopping them from giving a team like Calgary permission to speak with Chychrun prior to both sides calling in a trade.
With that said, Calgary may still prefer a Chychrun/Weegar swap even without talking to Chychrun. Weegar's NTC and term will make him more difficult to trade. If Calgary flips him for Chychrun, they have a more dynamic asset. They can either extend Chychrun, or they can trade him at the draft for a bigger return than Weegar would have gotten.
Despite what Dorion said in the past about Turris and Karlsson, there is nothing in the league rules that prevent management from talking to their own player prior to the date they are eligible to file an extension. If Ottawa allows Calgary to talk to Chychrun, both sides would have a pretty good idea about whether or not an extension would be feasible.
The other option would be a 3-way trade, or a proxy 3-way trade. Ottawa sends Chychrun to a team for a package of assets. Acquiring those assets then enables Ottawa to trade for Weegar.
If Chychrun is not going to extend, acquiring Weegar would be a best-case scenario. It's a shame, because the Senators probably could have had Weegar for cheaper than what they gave up for Chychrun had they traded Florida for him last summer, but you cannot go back in time...yet.