সে নদীর কাছে এক কুটিরে বাস করতো।

Explanation: Close by (something); 
English Meaning: nearby; adjacent; neighboring.
Bangla meaning: কাছে
Example sentence: The man lives close by the market.

তাই সঠিক ইংরেজি অনুবাদ হবে - He lived in a hut close by the river.

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Close to
Almost or nearly

Close with 
1: to approach close to.
2: to engage in hostile encounter at close quarters.
3: to ratify an agreement with.

Close at hand idiom
- Near in time or place

Source: Merriam-webster Dictionary, Collins Dictionary and Accessible Dictionary.