A little quick research shows that the 444 went through a number of changes during its short production history (1967 - 1971). They were manufactured in England and there were at least four different hydraulic pumps used. I don't know if there were any changes made to the accessory drive of the engine or not, but there very well may have been. There is also a difference between gas and diesel. It's looking like your project may be a bit more involved than first thought, but from what I found most of the pumps were from 4.5 to 9 GPM. Some of them used a tapered shaft for mounting the drive gear, and some used a straight shaft. If yours used a tapered shaft, it may be difficult to find an aftermarket pump that will work, as most of the off the shelf pumps use a straight shaft.
CaseIH still lists parts and complete pumps for all versions of that tractor, but I realize that you are trying to save some money and I cannot say as I blame you. Parts prices have been going through the roof over the last few years. I just put a new pump in a Massey Ferguson 2775. Glad we did the labor ourselves as the pump is located inside the rear end and is no fun to replace, so we saved some money there, but just the pump was $1200. It's an aluminum body and cannot be rebuilt so we hope it lasts a while! The Massey has a somewhat poor system: Hydraulic oil is also the lube oil for the entire rear half of the tractor and it is also shared with the power shift part of the transmission. If one thing goes wrong and contaminates the oil, the whole thing is in jeopardy.
Good luck in your search, hope you manage to find something.
ON EDIT: It appears that the pumps used are as follows:
GAS: 4.5 GPM for tractors with manual steering, and 9 GPM for tractors with power steering.
DIESEL: 7.6 GPM for tractors with manual steering, and 10 GPM for tractors with power steering.
It also appears that the mounting and drive gear are not the same for gas and diesel tractors.
Hope this helps!