He said: "It's looking like Sunday's game will come too soon for Afonso. We're now just waiting for him to get a visa so he can get back into the country.
"He's got international clearance, he's registered, he's our player and it's purely now about the fact that anybody coming to work in this country needs a working visa.
"We don't think he'll have that before the match but he'll certainly have it by the early part of next week. But this isn't just about one game, he's here for the long haul and there's plenty of time for him to make an impact.
"He hasn't played much football this year, so we don't want to rush things."
Boro smashed their long-standing transfer record to sign Alves for a reported £12million.
The club's previous record buy had been £8.15million Italian misfit Massimo Maccarone in June 2002, who came with a huge reputation but left on a free transfer early last year.
Unlike Maccarone - who had not played top-flight football when he left Italy - Alves arrives with a prolific strike rate of 45 goals in 38 games for Dutch Eredivisie side Heerenveen and eight Brazil caps to his name.
On finalising the deal, Southgate said: "It's fantastic news. I went down to London with Keith Lamb on Thursday for the work permit hearing and, to be honest, I never thought it was going to be a problem because of the quality of the player. He's the Dutch player of the year and he was the top scorer over there last season.
"When you think there are players of the calibre of Klaas-Jan Huntelaar in the Dutch league and Afonso was out-scoring him, that shows you the measure of his achievements.
"We are delighted to have brought such a player here. He was very keen to come here, he wants to play in the Premier League and he wants to score goals for Boro."