I know you said you were after something set in the present, but I'm going to recommend the Fallout series anyway. :P There are a few Sci-fi elements in it such as some energy-based weapons, but for the most part it seems to fit the criteria you posted - it's an open world game, you can recieve quests from various NPC's (as opposed to having them automatically given to you after meeting certain in-game conditions), there's no fantasy elements to speak of (as far as I'm aware) and the game's setting isn't *too* far ahead of our own time. Well, mostly. :P
Alpha Protocol dosen't feature an open world, but it's a modern-day RPG. The game got a mixed reception when it first came out, so it might be worth seeing what other people said about it or watching some gameplay videos first before you go for that one.
There aren't many others I can think of off the top of my head at the moment.
Edit: The Yakuza series would be a good choice too. None of the games feature open-world exploration to the same degree as the likes of Fallout, but the opportunity to explore is still there. Side quests are acquired by speaking to certain NPC's, and the series is devoid of anything fantastical.
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