The books make it clear that they offer the reverse service (see the following question: What Do the Gringotts Goblins Do With Muggle Money?).
According to the canonical answer to that question,
Q: When people trade in Muggle money for Wizard money, what does Gringotts do with the Muggle money?
JKR: Those goblins are sneaky people. They manage to put the Muggle money back into circulation. They are like "fences" --British slang, do you understand it? (src: America Online chat with JKR transcript, AOL.com, 19 October 2000)
The answer also makes reference to a comment by Dharini Chandrasekaran:
Just speculation, but squibs who go out into the muggle world to work will need Muggle money and also wizards might need it while doing Muggle studies research or trying to remain under cover, so I guess they keep it for reverse exchange.
This seems entirely reasonable speculation. Nevertheless, I don't recall any example in the books of converting Galleons to British pounds or other forms of Muggle currency. Does anyone have a more canonical source than speculation that indicates that this happens?
At one point in the books, Harry seems to overlook or ignore the possibility:
Stored in an underground vault at Gringotts in London was a small fortune that his parents had left him. Of course, it was only in the wizarding world that he had money; you couldn’t use Galleons, Sickles, and Knuts in Muggle shops. He had never mentioned his Gringotts bank account to the Dursleys; he didn’t think their horror of anything connected with magic would stretch to a large pile of gold. (CoS, Ch4, "At Flourish and Blotts"*)
(This passage was brought to my attention by DVK's answer here: Why didn't Harry Potter tell Aunt Petunia he was rich to improve how he was treated by Dursleys?)
But maybe he's just worried that if he converts some gold to Muggle money and tries to spend it, it will look suspicious if either Muggle authorities or the Dursleys find out.
Economically, I can't see any reason why they wouldn't offer this service. Galleons clearly have value, so the goblins should have no problem accepting them. They get some Muggle money from exchanges, and they seem resourceful enough to be able to obtain it in other ways if need be.
Converting wealth in the Wizarding world to wealth in the Muggle world is also the subject of the following question: Rich Wizards, Poor Muggles. However, I don't see an answer to my question there. One issue brought up is inflation of Muggle currencies, but as long as the goblins aren't printing Muggle money, I don't see why this would be a problem. There is also speculation that the Ministry of Magic doesn't want wizards to become suspiciously wealthy in the Muggle world.
Even if those answers are right, however, that doesn't answer my question. Gringotts could just restrict the amount of money that it will convert for a single person in a certain amount of time.
(In case it wasn't clear enough in previous versions of this question: I think that they do, but I wanted an answer that relies on more than speculation. Evidence could come from the books, direct or indirect, or from outside sources like author interviews, stuff like that. Economics in Harry Potter doesn't always seem to work the way I'd expect it to based purely on logic.)