Is it normal for a Professor to make unsolicited contact with potential Masters students via LinkedIn?

The Linkedin channel is unusual, but not suspicious per se. Does your friend have common connections with that professor? If that is the case, she might have popped up in the "people you may know" list that he gets. Or he did a keyword search. The one relevant thing with Linkedin is that he saw a — presumably flattering — picture of her.

He's told her that if she gets a good score on the GRE he will be able to admit her.

On that point again, I'd give him the benefit of the doubt. Maybe it's a wording issue and what was really meant was that with a good GRE score, admission in the local program is likely.

He has also made promises of grants.

This is rather standard for grad students to have their tuition and stipend covered by a grant.

keeping in touch via Skype

This, this. I have a hard time finding sound reasons for a professor to invest time and effort into communicating with a potential grad student, who did not even apply, by videoconferencing. This being said, Skype offers free international calls so it might be just for this reason.

So, there are sketchy aspects to that story, but nothing decisive. If the position is really of interest to her then maybe she should try investigating further. For example contacting the local graduate program admin and ask about the admission process, ask the alleged funding body for grant criteria, etc. in order to have multiple channels of communications. If she sees mismatches in the prof's story and theirs, that could give an additional warning.

I agree with a lot of the commenters, that this situation as you've described it raises red flags.

However, I do wonder if your friend will appreciate your involvement. Have you discussed this with her? There may be more going on here than you're able to describe to us, because you're not involved in the situation. If you've talked with her about your concerns and she seems receptive, then great. Otherwise I would tread very cautiously, particularly if you were thinking of using a SE post to make your point after she's disagreed with your assessment.

Regardless, I applaud you for your friendly concern.