"We've spoken with the club and the player but everything at the moment is a bit up in the air," Southgate told Boro's official website.
"What we've got to do is to negotiate with Heerenveen. He's on the open market so we can negotiate with them and go from there.
"If we can do it, it will give everyone a big lift. It shows ambition that's there from the chairman (Steve Gibson). It's a significant signal from him."
And Southgate has revealed that the two clubs have yet to agree a fee for Alves, who signed a contract with AZ before the Dutch FA (KNVB) ruled the Alkmaar club could not force Heerenveen to sell.
"Most of the parties are willing to do it but we have to agree a fee and it's not straightforward - unless we agree to their fee," Southgate added. "I don't want to spend money over and above what we can afford.
"His goalscoring record at Heerenveen is emphatic and he had a good record in Sweden before that. He is a goalscorer. Adapting to the Premier League is the challenge."