What you describe is called a conditional value list where the current client record needs to "filter" the list of possible contacts down to just those that are related to the current client. A basic conditional value list of this type is usually set up with a relationship that matches only to records that make up the list of values you want to see for a given client and a "use values from field" value list that specifies the "include only related values" option.
I suggest taking a look at "Adventures in FileMaking #1 - Conditional Value Lists". This demo file contains detailed descriptions of actual working conditional value lists and demonstrates 10 different ways to set up a conditional value list. The "Basic" conditional value list sounds exactly like what you need here.