Ramirez confirmed that an agreement was in place with the Saints for what would be a club record fee of £11.8m earlier today.
Adkins, who has already seen the club's transfer record broken once this summer with the signing of Jay Rodriguez, described the possibility of the 21-year-old Uruguay international coming to St. Mary's as "exciting".
"It looks exciting, doesn't it? I think he has come out in Italy and said it is already sorted, but I think you'll find the policy of the football club is until the player has signed on the dotted line and we announce it then really we try to do everything privately," said Adkins.
"We have been consistent with that over the years and we'll probably stay that way. If it is true, then it obviously shows the ambition of the football club. I know the chairman, Nicola Cortese, is very ambitious here and it is great. We were in administration a few years ago and we've come a hell of a long way in a very short space of time.
"To be linked with players as such - and don't forget we broke the (club) transfer record in the summer when we signed Jay Rodriguez - shows we're very ambitious, but likewise we'll cut our cloth accordingly. Having been in administration once, the last thing we'll do is end up in that situation again. The club is run very well, very prudently. But we're very, very ambitious and having reached the Premier League now, very quickly, we've got to give ourselves an opportunity to do well there."
Ramirez has also been linked with the likes of Tottenham Hotspur, Liverpool and Inter Milan this summer.