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I need an approach to combine points close one to another into a common point.

I can take an average of points' {lat, long} to create one point.

How I can do this with ArcMap or Python? I have an Excel file for those points.

Should I combine according to {lat, long} information with Excel?

Example

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My interpretation of your question is that you are not simply trying to set a zoom level on an interactive map, and your point groups near each other do not share a common attribute. If my understanding of your question is incorrect, please modify your question to provide more details, as you will get a different answer.

Based on my two assumptions, and piggy-backing off the earlier suggestion to use density-based clustering, here is what I would try using Arcmap (after converting your Excel file into a feature class):

  1. Determine the clusters using the Grouping Analysis tool. Use the x and/or y values already present in your dataset, and try different values for the various spatial parameters. This will result in a new feature class with a column that identifies which group each feature belongs to. Symbolize this layer by group number to determine if you are happy with the groupings.
  2. Dissolve the points to create multi-part features based on the group number assigned in Step 1.
  3. Use Feature to Point to obtain the centroid of each multipoint cluster.

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