First remember the consultant works for the company after all you are not paying their salary.
Getting an offer letter depends on several factors such as, but not limited to,
-- Just how urgently the company wants to fill the position - although saying that they need someone ASAP could be reality although I have known some companies who use it as a tactic to get people to let their guard down and jump.
-- The company could have had an unexpected event happen which required their immediate attention and filling the role has lost priority
-- The company could also continuing interviewing other people or the consultant could still be presenting possible candidates to the company -- telling you an offer is in the works so that if one does surface you can jump and the consultant can get paid
The possibilities are endless.
I would suggest that you continue to interviewing as nothing is definite until all parties have signed then there still is the probationary period to go through.
Also, once you have the offer letter then you officially begin the negotiation process (better salary, more vacation days, more frequent performance/salary review, bonus, medical, living allowance, etc.)
Best of Luck!