It was very possible, since the "Southern" capital was made a fait accompli via a deal. Jefferson and his southern allies accepted Federal assumption of the states' remaining war debts, and Hamilton and his northern allies backed a capital on the Potomac. (The debt assumption was particularly troublesome for Virginia and other states that had already paid off their debts, since federalisation meant that the remaing war debts of Northern states would then be collectivised. Virginia and others would thus pay twice, also carrying part of the burden for New York at al.)
If the deal falls through, it's possible to see New York as the site for the Federal capital. The question is whether Manhattan is the logical option. Manhattan was NYC at the time, and ceding it would be a big loss for the state of New York. Furthermore, it would leave Long Island and Staten Island pretty much split from New York state. (It's conceivable that staen Island would seek to become part of New Jersey, and Long Island might wish to become a separate state, in a situation not entirely unlike Maine-- which was initially part of Massachusetts.) I'm not sure the state of New York would be happy about having to deal with all these matters...
An alternative might be to put the Federal district on Staten Island, whih makes infinitely more geographical sense, and mirrors the OTL notion of "district between Virginia and Maryland" by putting the district between New York and New Jersey. The main objection would be that Staten Island had been a loyalist hotbed during the war, and George Washington once said something like "the Staten Islanders are our fiercest enemies". Then again, most loyalists had fled after the war, and putting your Federal district in their former hotbed might make for a great 'fuck you!' to them.