At least in bacteria they can. In optimal conditions, E. coli divides itself about once every 20-30 minutes. Wich is more important, the replication time of its genome is about 40-50 minutes. That means that, in exponential phase, E. coli has at least two active replication bubbles. Since the density of ribosomes must be more or less constant, aswell that the new cells needs to synthetise peptidoglycan and other cell components, that means that replication and transcription occurs simultanously. If not, ribosomes, enzimes, tRNAs and mRNAs would be diluded in every replication event.
In eukarya I see no reason of why that shouldn't be possible. The formation of chromatin is not a random event and some specific proteins are required during short periods of time during some steps of mitosis, so it's logic to assume that some specific parts are not compacted or that transcriptional activity can occur even in chromatin.
The specific region that is duplicating itself in a given time will probably be unable to transcribe, because the polymerase complex is pretty big and they need to open the double helix, and that. But this is somehow like children playing in the train tracks. Once the train aproaches, the children stay apart, and they continue again when the train has gone.