The correct answer is D. They told him that they would visit his house the next day.
The sentence “They said to him, ‘We shall visit your house tomorrow'” is a direct statement of intent. The use of the word “shall” indicates that the speaker is making a promise or commitment to visit the listener’s house on the following day. The other options are incorrect because they do not convey the same level of certainty or commitment.
Option A is incorrect because it uses the word “will,” which is a less formal and less certain way of expressing intent. Option B is incorrect because it uses the word “desire,” which suggests that the speaker is not sure whether or not they will actually visit the listener’s house. Option C is incorrect because it uses the word “might,” which suggests that the speaker is only considering visiting the listener’s house and has not made a firm decision.