In a sense, this is a strictly hypothetical question, since the Bible clearly teaches that Jesus never sinned (Hebrews 4:15). However, theologians have discussed it at length. In technical terms, "impeccability" is the doctrine that Christ could not sin, though he could be (and was) tempted.
Christian churches generally agree that:

Jesus never sinned, but he was tempted.
"For we do not have a high priest who cannot sympathize with our weaknesses, but one who has been tempted in all things as we are, yet without sin." (Heb. 4:15 NASB)

Any temptations he faced were directed at His human nature, rather than His divine nature.
"Let no one say when he is tempted, 'I am being tempted by God'; for God cannot be tempted by evil, and He Himself does not tempt any one." (James 1:13 NASB)

Some theologians, such as Charles Hodge, have argued that Christ could have sinned. Their main argument is that temptation is not real unless there is a possibility of yielding to it.
Others (with whom I agree) make the following case for the idea that Jesus could not have sinned.

It is possible to attempt the impossible, and temptation is merely Satan's attempt to lead us astray. Temptation and susceptibility do not have to go together; only successful temptation always goes hand in hand with susceptibility.

All the temptations Jesus faced in Matthew 4 were legitimate offers. The struggle in Gethsemane was a real, difficult experience. One might even say that Jesus experienced worse temptation than we do, because we usually surrender before the devil brings out his worst weapons. Jesus stayed firm through the most powerful temptations in Satan's arsenal.

Although the human nature of Jesus may have had desires which temptation could target, His divine nature would ensure that He would never actually decide to sin. It is impractical to speculate on what Christ's human will might have decided if it were operating independently of His divine nature; such a situation never arose.

Christ's divine attributes would make it impossible for him to sin:

Immutability - His basic nature never changes (Hebrews 1:12; 13:8). He was holy in eternity past, and He would remain holy now.

Omnipotence - Falling to temptation shows moral weakness or lack of power and ability. Christ had infinite power, and was therefore not susceptible to sin.

Omniscience - Satan tempts us by attempting to deceive us. Jesus had infinite knowledge, so he could not be deceived.