The US already has a presence in many locations in Scandinavia/The Nordics. Depending on what one considers to be a base, as many as 36.

In total, Norway, Sweden, and Finland have opened 36 military bases for US forces and weapons. The agreements are bilateral, i.e., between the US and the individual country, and not a NATO agreement.

But yeah, more will undoubtedly not make Putin any happier.

Source.