The text mentions nothing that Prince Prospero actually "offers" to the specter, unless you are refering to the way in which Prince Prospero in a frenzy of shame and anger at his cowardice pursues the mummer with a dagger in his hand, planning to drive the dagger deep into the mummer's body for the insult that he has caused having dressed up as the Red Death. Note what the text tells us about this:
It was then, however, that the Prince Prospero, maddening with rage and the shame of his own momentary cowardice, rushed hurriedly through the six chambers, while none followed him on account of a deadly terror that had seized upon all. He bore aloft a drawn dagger, and had approached, in rapid impetuosity, to within three of four feet of the retreating figure...
Thus we could argue that Prince Prospero "offers" the person of the mummer a drawn dagger as he pursues him, if that is what your question is refering to. I hope this answers your question. If not, please do respond and perhaps see if you can clarify what it is precisely that you want to know.