This sounded like an interesting book to read, so I did some extensive digging. Both Asimov and Clarke were very visible and popular figures, and their careers and writings have been extensively documented.
I started with the ISFDb pages for Isaac Asimov and Arthur C. Clarke. These are extensive and apparently quite complete for their genre writings. There is even an entry on Clarke's listing for an untitled note of congratulations to Asimov. But notably there are no listings for any collaborative works. There is also no mention of any collaboration in either's Wikipedia biography (Asimov, Clarke) or bibliography (Asimov, Clarke) nor in their respective entries in the SF Encyclopedia (Asimov, Clarke). Other resources (e.g. Asimov Online, worldcat, Internet Archive) also fail to find anything.
There are 2 books listed for Clarke which are records of correspondence, one with Peter Hyams and one with C.S. Lewis, so if such a book existed it should definitely appear.
I decided to approach it from the other end. A story told as an exchange of letters/messages is called an "epistolary novel." (List entry) There is a TVTropes entry for them, which lists 2 stories by Asimov ("Blind Alley" and "Rejection Slips") but nothing by Clarke and no collaboration.
I searched a couple of online resources (questia, worldcat) for books about both Asimov and Clarke, but again, didn't find anything that seemed relevant.
Ultimately I have only 2 very loose possibilities for matches:
Yours, Isaac Asimov: A Lifetime of Letters (1995) contains a chapter "Clarke, de Camp and the del Reys" which in addition to letters from Asimov includes 3 letters from Clarke, but it isn't a story, and Clarke's letters aren't a significant contribution to the overall work.
Finally, the best I can come up with is Space Mail, Volume II (1982) which is an anthology edited by Asimov (et al.) containing stories which are all told in epistolary form. One of these stories is "Loophole" by Arthur C. Clarke. If nothing else it's in the right time period.