diff --git a/src/Storages/MergeTree/MergeTreeDataSelectExecutor.cpp b/src/Storages/MergeTree/MergeTreeDataSelectExecutor.cpp
index 98a08abab654a0556b1abc31b45dd8d19585d2a9..58193ec040fde844125ba2da991f69cd0f1491ba 100644
--- a/src/Storages/MergeTree/MergeTreeDataSelectExecutor.cpp
+++ b/src/Storages/MergeTree/MergeTreeDataSelectExecutor.cpp
@@ -833,15 +833,14 @@ QueryPlanPtr MergeTreeDataSelectExecutor::readFromParts(
 namespace
 {
 
+/// Marks are placed whenever threshold on rows or bytes is met.
+/// So we have to return the number of marks on whatever estimate is higher - by rows or by bytes.
 size_t roundRowsOrBytesToMarks(
     size_t rows_setting,
     size_t bytes_setting,
     size_t rows_granularity,
     size_t bytes_granularity)
 {
-    /// Marks are placed whenever threshold on rows or bytes is met.
-    /// So we have to return the number of marks on whatever estimate is higher - by rows or by bytes.
-
     size_t res = (rows_setting + rows_granularity - 1) / rows_granularity;
 
     if (bytes_granularity == 0)
@@ -849,6 +848,34 @@ size_t roundRowsOrBytesToMarks(
     else
         return std::max(res, (bytes_setting + bytes_granularity - 1) / bytes_granularity);
 }
+/// Same as roundRowsOrBytesToMarks() but do not return more then max_marks
+size_t minMarksForConcurrentRead(
+    size_t rows_setting,
+    size_t bytes_setting,
+    size_t rows_granularity,
+    size_t bytes_granularity,
+    size_t max_marks)
+{
+    size_t marks = 1;
+
+    if (rows_setting + rows_granularity <= rows_setting) /// overflow
+        marks = max_marks;
+    else if (rows_setting)
+        marks = (rows_setting + rows_granularity - 1) / rows_granularity;
+
+    if (bytes_granularity == 0)
+        return marks;
+    else
+    {
+        /// Overflow
+        if (bytes_setting + bytes_granularity <= bytes_setting) /// overflow
+            return max_marks;
+        if (bytes_setting)
+            return std::max(marks, (bytes_setting + bytes_granularity - 1) / bytes_granularity);
+        else
+            return marks;
+    }
+}
 
 }
 
@@ -904,11 +931,12 @@ QueryPlanPtr MergeTreeDataSelectExecutor::spreadMarkRangesAmongStreams(
         data_settings->index_granularity,
         index_granularity_bytes);
 
-    const size_t min_marks_for_concurrent_read = roundRowsOrBytesToMarks(
+    const size_t min_marks_for_concurrent_read = minMarksForConcurrentRead(
         settings.merge_tree_min_rows_for_concurrent_read,
         settings.merge_tree_min_bytes_for_concurrent_read,
         data_settings->index_granularity,
-        index_granularity_bytes);
+        index_granularity_bytes,
+        sum_marks);
 
     if (sum_marks > max_marks_to_use_cache)
         use_uncompressed_cache = false;
@@ -1035,11 +1063,12 @@ QueryPlanPtr MergeTreeDataSelectExecutor::spreadMarkRangesAmongStreamsWithOrder(
         data_settings->index_granularity,
         index_granularity_bytes);
 
-    const size_t min_marks_for_concurrent_read = roundRowsOrBytesToMarks(
+    const size_t min_marks_for_concurrent_read = minMarksForConcurrentRead(
         settings.merge_tree_min_rows_for_concurrent_read,
         settings.merge_tree_min_bytes_for_concurrent_read,
         data_settings->index_granularity,
-        index_granularity_bytes);
+        index_granularity_bytes,
+        sum_marks);
 
     if (sum_marks > max_marks_to_use_cache)
         use_uncompressed_cache = false;
diff --git a/tests/queries/0_stateless/01665_merge_tree_min_for_concurrent_read.reference b/tests/queries/0_stateless/01665_merge_tree_min_for_concurrent_read.reference
new file mode 100644
index 0000000000000000000000000000000000000000..6ed281c757a969ffe22f3dcfa5830c532479c726
--- /dev/null
+++ b/tests/queries/0_stateless/01665_merge_tree_min_for_concurrent_read.reference
@@ -0,0 +1,2 @@
+1
+1
diff --git a/tests/queries/0_stateless/01665_merge_tree_min_for_concurrent_read.sql b/tests/queries/0_stateless/01665_merge_tree_min_for_concurrent_read.sql
new file mode 100644
index 0000000000000000000000000000000000000000..ca324acdce353dff23ef67eab57d122d454fda9f
--- /dev/null
+++ b/tests/queries/0_stateless/01665_merge_tree_min_for_concurrent_read.sql
@@ -0,0 +1,6 @@
+DROP TABLE IF EXISTS data_01655;
+CREATE TABLE data_01655 (key Int) Engine=MergeTree() ORDER BY key;
+INSERT INTO data_01655 VALUES (1);
+SELECT * FROM data_01655 SETTINGS merge_tree_min_rows_for_concurrent_read=0, merge_tree_min_bytes_for_concurrent_read=0;
+-- UINT64_MAX
+SELECT * FROM data_01655 SETTINGS merge_tree_min_rows_for_concurrent_read=18446744073709551615, merge_tree_min_bytes_for_concurrent_read=18446744073709551615;