Here are some rough numbers. This isn’t intended as a comprehensive analysis but as something a bit more accurate than a SWAG (Scientific Wild Ass Guess).

Let’s start with the STS-107 info from the Encyclopedia Astronautica website:

16 January 2003 15:39 GMT. Duration: 15.94 days. Call Sign: Columbia. Nation: USA. Agency: NASA. Launch Site: Cape Canaveral . Launch Complex: LC39A. Launch Vehicle: Shuttle. LV Configuration: STS-107. Program: STS. Class: Manned. Type: Spaceplane. Spacecraft: Columbia. Payload: Columbia F28 / Spacehab. Perigee: 280 km. Apogee: 280 km. Inclination: 39.0 deg.

Now, we need the basic Shuttle Orbiter numbers, again from the Encyclopedia Astronautica:

Gross Mass: 99,318 kg. Empty Mass: 99,117 kg. Thrust (vac): 696,905 kgf. Isp: 455 sec. Burn time: 480 sec. Propellants: Lox/LH2 Isp(sl): 363 sec. Diameter: 4.9 m. Span: 23.8 m. Length: 37.2 m. Country: USA. No Engines: 3. SSME Status: Hardware.

Since there is no propellant available to power the SSMEs once the ET is released, any significant maneuvering in space is done using the OMS engines. The specs for the OMS is listed as:

Gross Mass: 25,200 kg. Empty Mass: 3,600 kg. Thrust (vac): 5,443 kgf. Isp: 316 sec. Burn time: 1,250 sec. Propellants: N2O4/MMH Isp(sl): 0 sec. Diameter: 4.9 m. Span: 23.8 m. Length: 37.2 m. Country: USA. No Engines: 2.

So, in round numbers, the on-orbit mass was approximately 100,000 kg and the Isp for the OMS engines is 316 seconds. At launch, the OMS system has approximately 21,600 kg of propellant. Some of that is used for the orbital insertion burn, some is used for the attitude control thrusters, some is reserved for the deorbit burn and for the thrusters during reentry, and the rest is available for maneuvering.

The basic numbers for the ISS orbit is an inclination of 50.5 degrees and an altitude of approximately 390 kilometers. The altitude varies over time due to slow orbital decay and thruster firings to boost the station, but let’s use 390 kilometers for simplicity. I honestly don’t know the ISS’s altitude during the Columbia mission.

So, we have a difference in inclination of 11.5 degrees and a difference of altitude of about 110 kilometers. In the best case, you can combine the firings to change the inclination and altitude together. However, the situation may be a lot more complicated than that. In addition to the basic altitude and inclination, you have to consider other factors like the Right Ascension of the Ascending Node (RAAN). I don’t have the orbital parameters handy for the ISS and Columbia during the STS-107 mission. In the best possible case, the RAAN for the two orbits would be close. In the more likely case, the orbital planes weren’t all that close. In order to rendezvous, both vehicles have to be in exactly the same orbit.

Still, let’s keep it simple. Let’s just assume that all the Columbia had to do was change the inclination by 11.5 degrees and change altitude by 110 kilometers. We’ll ignore everything else like the probable difference in orbital plane. This would be the best possible situation and would be unlikely to ever happen in the real world.

First, according to this source, if you spent all of the Shuttle’s OMS propellant for a plane change, you could only change the inclination by approximately 1 degree.

Let’s start running some numbers beginning with the orbital velocities at 280 (Columbia) and 390 (ISS) kilometers using this calculator:

The orbital velocity at 280.00 kilometers is 7.74 km/sec! The period of the orbit is 90.11 minutes.
The orbital velocity at 390.00 kilometers is 7.67 km/sec! The period of the orbit is 92.36 minutes.

Now, we need to change both the altitude and the plane, as discussed here.

Now, we need to calculate how much Delta-v would be required to change altitude from 280 to 390 kilometers. According to this handy calculator, it takes a Delta-V of 0.700 kilometers (remember to add in the Earth’s radius of 6378 kilometers).

Then there’s the Delta-V (or Delta-I) necessary to change the inclination from 39 degrees to 50.5 degrees. According to this calculator, changing the inclination at 390 kilometers would take 1.53 km/sec change in velocity.

A combined maneuver could probably require a total Delta-V of 1.7 km/sec from the square root of (.49^2 + 1.53^2).

Now, we need to determine how much propellant would be necessary to impart a 1.7 km/sec Delta-V in a vehicle with a mass of approximately 100,000 KG using rockets with an Isp of 316 seconds. Well, this rocket equation calculator indicates that it would take a mass ratio of 1.73 (empty mass to fueled mass) to do this. That implies that it would’ve taken Columbia about 73,000 KG of OMS propellant to rendezvous with the ISS under the best possible circumstances. That’s over 50,000 kg more propellant than Columbia had at liftoff.

Actually, that sounds kind of low to me. Perhaps I'm missing something. Still, it shows in general terms how impossible it would've been for Columbia to dock with the ISS even under the best of circumstances.