It seems unlikely that if they offered asylum they would then reject it after he arrived in the country. An application can not hurt and makes it likely for the country to make a public statement.

We do have a constitutional position of neutrality, which may underpin his appeal. But since he's not fleeing an armed conflict I don't see this availing him particularly.

Seems like most of Europe will require asylum applications to be left in person inside the country. That will make Snowdens life bit more difficult, especially as his mobility seems bit limited atm.