My admittedly speculative answer is no. Beyond the longer term wisdom of doing so or not, I've noticed two things in the last couple of days that point me to this conclusion:

1. The Bulls have shown no sign to me that they are looking to exceed the luxury tax threshold. As currently constructed, the Bulls can fill out their roster with minimum salary guys and still avoid the tax. But if they match Omer's offer, they'll pay the tax. But... if they were willing to pay the tax, it stands to reason to me that they would have been quite a bit more aggressive in at least looking at other means of improving the team. As it stands, the only thing they were rumored to be interested in doing was a Luol Deng for lottery pick + salary dump sort of deal.

2. I'm not 100% sure on this, but did anyone else notice the way the Pacers "matched" Roy Hibbert's offer today? Per David Aldridge

Indy, per source, will give Hibbert the same terms that Portland was ready to give in offer sheet, & so won't technically "match" the sheet.

Does this mean that, since the FA signing process hasn't officially gotten under way, and no offer sheets have officially been signed, none of this is binding? That is, the Rockets and Asik have declared their intent, but it isn't set in stone until an offer sheet is signed... and the Bulls could simply match the offer before then. This seems a bit at odds with the general way the CBA operates, but it's important for a couple reasons. First... the Bulls have been very silent. If they were gonna match, why not just say so. Second, a big problem with the Rockets offer is the poison pill third year at $14-15M. But if that offer hasn't actually been signed and the Bulls are free to negotiate, they could simply negotiate around it and offer Asik something like a straight $8M per year. If they so chose. I take it from the fact we've heard nothing about it that they do not so choose.