WASHINGTON — The Supreme Court signaled on Tuesday that it was struggling with two conflicting impulses in considering a request from television broadcasters to shut down Aereo, an Internet start-up they say threatens the economic viability of their businesses.

On the one hand, most of the justices suggested that the service was too clever by half, with a business that relies on capturing broadcast signals and streaming them to subscribers for a fee.

“Your technological model,” Chief Justice John G. Roberts Jr. told Aereo’s lawyer, “is based solely on circumventing legal prohibitions that you don’t want to comply with.”

But the justices were also clearly concerned with the impact that a ruling against Aereo could have on future technological innovation.