He started last season again as the only senior keeper, but was sidelined for a month after suffering a fractured cheekbone in the 0-0 draw with Queens Park Rangers in October.
Former Greece Under-21 international Dimi Konstantopoulos was brought in as cover and almost joined permanently to challenge De Vries for his place, but missed the transfer deadline, leaving the Dutchman finishing the season as the only keeper.
"I've always said that I wanted to stay at Swansea because I know the set-up has been magnificent under Roberto Martinez," said De Vries.
"Things have now changed at Swansea and let's hope things will go in a positive way.
"I think the chairman has made a good step by appointing Paulo Sousa as the new manager."
De Vries added: "Paulo said he would like to work with me as the goalkeeper at Swansea and everything was positive.
"It's up to the chairman [Huw Jenkins] and my agent to get things sorted and hopefully it will be in a positive way.
"Pre-season training starts on Monday and I think the chairman wants it to get sorted before then.
"Everything can be really quick if things go well and let's hope they do."
Jenkins said: '' We are all very keen to get him signed up, we will see what we can do over the next couple of days.''
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