Both are possible.
Hello, Teacher, are both sentences correct?
(1) I spoke with Mr. Lin today to enquire when he could meet with the Manager. He said the only date he is available next week is April 26th.
(2) I spoke with Mr. Lin today to enquire when he could come meet with the Manager. He said the only date he was available next week was April 26th.
Both are possible.
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I prefer #1:
(1) I spoke with Mr. Lin today to enquire when he could meet with the Manager. He said the only date he is available next week is April 26th.
I would be more likely to use "is" here simply because we haven't reached April 26th yet. If, however, I were talking about a conversation which happened in the past and the date that the manager had specified is now also in the past, I would probably use "was".
I spoke to the manager. He said (or "says" in BrE) the only date he is available next week is April 26th,
I spoke to the manager. He said the only date he was available the next/following week was April 12th.