The left-handed batsman has agreed a short-term two-and-a-half-week contract to join the reigning county champions, but hinted that he is open to a more long-term return in future.
"If I can have another year here, where I can be successful, I would love that," said Sangakkara, who helped his country to lift the World T20 crown earlier this year.
"The real key is the relationships that you build up over time. Durham were one of the few counties with an opening for an overseas player.
Sangakkara will make his Durham debut in the four-day game against Yorkshire in the County Championship on Sunday. His second and final game is against Sussex at Hove, starting the following weekend.
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The short-term stay provides ideal preparation for Sri Lanka's forthcoming Test and one-day international series against England, which starts at the end of May.
The 36-year-old Sangakkara, who is ninth on the list of all-time Test run scorers with 11,151 at an average of more than 58, added: "It's important that I keep my focus on the present, and make an impact against Yorkshire.
"We have guys who can swing and seam, and others pushing into 140 kilometres per hour. Back in 2006, when we toured here, it was a pretty successful time, drawing the Test series 1-1 and winning 5-0 in the one-dayers.