a. The near future
b. The near house
Would you tell me which one is right?
As far as I know, instead of b, we must say: the close by/nearby house
Am I right?
Using "near house" here is wrong and the other sentence is correct. In both cases "near" is used as an adjective.
In near future
Here "near" = only a short time ahead
Near - Located a short distance away.
Example - The shop near the house is a small one.
But you can't say "the near house". You are correct to replace "near" with "nearby". If we say "the near house", we haven't provided any reference point to explain from what the house is "near".
Both possibilities are correct. Here are some examples of use for "the near house":
"In the picture, the far house on the corner was built by Henry Miner, and the near house, which has retained its original porch, was built by Horace Miner" (Early houses in Berlin, Wisconsin, and the people who lived in them - Lulubelle C. Gillett, 1976)
"The superstructure of the near house was completely demolished" (Pit and Quarry - Volume 45 - Page 165)
"A 16 by 16-foot office built of prefabricated panels has been added at one end of the near house" (Farm Business News - Volumes 39-40 - Page 51)
And here some examples for "the near future":
"We care more about the near future even in the special cases in which we cannot affect it"
"Russia simply does not have the economic means to redominate the entire Central Asia, at least in the near future" (Hooman Peimani - 1998)
"In the near future, billions of entities will be connected to each other through the Internet"
Although both, "the near future" and "the near house", are possible, not all of them are used with the same frequency.
By comparing how frequently these collocations are used, it is possible to understand why the other answers recommend the use of "the nearby house" over the "the near house".
One definition of nearby is close by, so when referring to relative locations (in this case you and the house) nearby is the best choice.
Also, near can be used for future points in time, but nearby cannot.