what exactly are the system specs...
running close to max will cause heat, random shut downs, and possible damage to other parts of the system. just because your systems psu is rated at 375 watts doesnt mean you have 375 to use. that may be the peak output and may not be stable. resulting in powerspikes and other potentialy harmfull issues.
The HD 4670 will be able to run on an average 375w PSU assuming you do not have many components that draws a lot of power like having 8 hard drives and an overclocked Core i7-2600K CPU.
It's best to list your components and precisely which 375w PSU you have.
If you have the $$$ for a Radeon HD 5670 (which is probably around $60 - $70 after rebates), then that would be a better choice since it uses less power than the HD 4670 and is also gives on average around 25% - 30% better performance in games.
There are programs on the Internet that allow you to input all your computer details and they tell you the WATTAGE you need. Some programs are more up to date and/or better than others.
To be clear, you should know:
1) the approximate Wattage that you need
2) the Amps you need for your graphics card (get at least 25% better than your graphics card rating. For example if the card needs 10A, then 10x1.25 is 12.5Amps minimum on the single or combined +12V rail/rails.
3) connection types, such as 6-pin or 8-pin which are on the graphics card but which might not be on the PSU.