Could you possibly lend me some money? is less direct than Lend me some money.
Let's see some other examples on how to soften what we want to say:
Instead of DO YOU MIND IF I OPEN THE WINDOW? use WOULD YOU MIND IF I OPENED THE WINDOW?
Instead of I WANT TO SPEAK TO YOU use I WANTED TO HAVE A WORD WITH YOU, IF THAT'S ALL RIGHT.
Observe these other sentences and check how the degree of formality varies:
CAN YOU HELP ME?
COULD YOU HELP ME?
I WONDER IF YOU COULD HELP ME.
I WAS WONDERING IF YOU COULD POSSIBLY HELP ME.
...
IT'S A BAD IDEA.
I'D SAY IT'S A BAD IDEA.
I'D HAVE SAID IT WAS A BAD IDEA IF YOU HAD ASKED ME.