This is what happens when you are talking to approach control and your flight path takes you into Class C airspace. Given you've already told the controller your destination of course, the approach controller will get an approved "point out" from the ATC facility which in this case would be the tower. It can be automated (the approach controller pushes some button on their keyboard and your data block flashes (yellow in my case, no idea of what other colors they might use elsewhere) which is shown on the tower's radar screen. The tower controller will accept the flash on their end OR they will actually unable the flash via a keystroke on their keyboard, OR they will actually talk to the approach controller via the shout line and find out exactly what you are doing before accepting the point out. In either case, YOU are not required to talk to the tower controller for entry into their Class C airspace, that is already been taken care of by the approach controller. Pretty much the same for Class D airspace as well provided they have a radar feed in their tower, otherwise it will be accomplished via the shout line to approach control.
This is accomplished every day, several times a day and neither facility is ever "surprised" that you are going to fly through their airspace. Its all there on the scope, callsign, type aircraft, speed, altitude, destination. This is NOT a big deal.