So, my players want the NPCs to communicate with one another more, and I don't know how to handle that without taking spotlight away from the PCs... Anything I can do to compromise?
NPCs can talk to each other without talking to each other.
You cool your heels in the administrative annex for a bit and listen to the conversation drifting in from the break room about the new food truck on Fourth Street that rapidly turns into a compare-and-contrast session about lunch.
The receptionist is patiently detailing the need to schedule a conference room for the third time to either one extremely forgetful person or an entire phone chain who don't explain anything to each other.
Through your boss's window you see a very animated discussion around a map of Pakistan that, yes, has finally reached its fifth color of marker.
I find it far easier to describe what a conversation is about than to actually come up with the content of the conversation, and honestly, describing what a conversation's about is probably going to save a lot of other people's time as well. For purposes of establishing scene texture and sketching out character, the words people say are very seldom as important as the way they say them.
Your players are likely looking for a world with consequence
What that means is that they feel like everything in the world revolves around them. They're looking for stuff to happen in the background. For example, the head of the thieves guild sends them on a mission to retrieve a special gem from a rival guild. While your group is doing this, the rival guild strikes out and topples the guild you're working for. When you return, you find said guildmaster about to be beheaded, and you're greeted by the new head of the thieves guild who tells you he'll be with you in just a minute, and then proceeds to dispatch his rival.
He then compliments you on your successful theft, having surmised this would be an opportune moment to seize control as the guild that sent you would rightfully expect that your group was distracting this guild. They miscalculated, and the new head of the guild considers the gem you retrieved as payment for essentially opening a vulnerability in the guild he wanted to control.
This is NPC vs NPC interaction, shows consequences in your realm, and will likely add some meat to your games that your players are seeking.
If you are referring to NPCs that are in constant contact with the PCs you could create a situation between the NPCs only a PC can resolve, or obligates the PC to step in to prevent an unwanted outcome.
This could mean settling a dispute or answering a gap in knowledge. Perhaps one NPC is failing at cheering up the other and the PC can step in and help. Two NPCs are hatching a plan doomed to failure and will have disasterous consequences for the PCs if they don't step in. One NPC busts in the middle of your discussion to tell you how whatever you are doing less important that the news he has to share right now.
Another option is to have a leader/subordinate NPC-NPC relationship, where one directs the other, or a hero/sidekick relationship like Sherlock Holmes and Watson.
You could also try roleplaying the conversations in full... but with the players running the involved NPCs. This won't work for groups where players want to be fully immersed in one, and only one, character, but I've had great success with it in groups which are even slightly flexible (because I still only have them playing a single character at a time, it just isn't always the same character) about that sort of thing.
My most common approach for this is to hand out slips of paper with a few pieces of information that each NPC knows, dividing it all up so that (even if the NPCs already know everything) no one player receives all of the relevant information. Then the players play out the conversation as the NPCs, working in the information they have, until they've learned the full picture.