Should I try to remove thermal 'paste' around chipset die?

[Car712]

Hey, so I am fixing and cleaning up an old Dell Dimension for someone else, and I decided to check the chipset to see if I could replace the thermal paste on it, well, clearly it needed it so I cleaned the old paste off the heatsink with a bit of work, but there is still a fair bit of it dried around the chipset die itself. I'm not really sure what to do here since I don't want to possibly damage it, but it would be great if I could clean up the old paste.. Heres a pic of the chipset:

[Fasauceome]

all that matters is that the top of the die is clean. You could take a cue tip and wipe off the excess on the sides if you really wanted to, but it won't affect anything.

[thatpc.]

You might want to replace it if you can because I had a old optiplex 960 and it’s southbridge got pretty hot sometimes. As @Fasauceomesaid the top of die needs to be clean but the sides don’t matter that much