The fan video What if "Star Wars Episode I" Was Good suggests that the script might have been improved if Palpatine was not a senator, but rather the full-time liaison between the Jedi Council and the Senate (thus allowing him to personally know each of the Jedi and be able to keep a closer eye on their actions). However, I don't see any evidence that this sort of position actually exists in canon. This made me wonder; who has the official authority to talk with the Jedi Council on behalf of the Senate (or vice versa), and how often?
In The Phantom Menace, we see that the Jedi Council and the Senate really don't talk too much. We know from the opening crawl that "the Supreme Chancellor has secretly dispatched two Jedi Knights" to investigate the issues on Naboo, so he is at least in communication with the Jedi Council. For the rest of the movie, the Jedi Council is really only involved with the Senate because they are already involved with Queen Amidala. I'm not sure if there were other communication channels between the Senate and the Jedi Council that were described in non-movie media.
So how much official communication was there between the Senate and the Jedi Council? Who from the Senate had the authority to talk with them and how frequent was that communication?
Legends information is very much welcome, but any answers with Legends info should also contain Disney canon information.