Is it possible to use an external monitor on an iMac (2010 intel i7 27 inch) that the graphics card isn't working

1) The 2010 iMac is not compatible with newer late model 27" external displays.

2) You would need to buy and use a Mini DisplayPort to VGA or DVI adapter.

3) You would need to buy or have a VGA or DVI equipped external display.

4) Working would depend on what is actually wrong with the 2010 iMac.

IMHO you would be far better off replacing older programs with new ones and getting an external display for the MacBook Pro, instead of wasting any time or money on a 12 year old iMac. see > Use an external display with your MacBook Pro - Apple Support

If the gfx card is not working, no external monitor will work either.

There are used/pulled gfx cards out there but the better ones tend to cost several times the realistic selling price of a used 2010 iMac. Also, you need some computer hardware skills of at least a stiff drink to remove and replace the card. A pro will likely charge 1 to 1.5 hours labor min at ~US$75-90/hour.

FWIW, my 2010 iMac 27" 2.93ghz i7 died the same way in late 2018. Although I am comfortable working inside computers, I bit the proverbial bullet and bought a new iMac rather that lose work time shopping for and installing a new card.

If the graphics card isn't working I'm not sure that you would be getting anything displayed.

You're welcome.