diff --git a/r/inst/build_arrow_static.sh b/r/inst/build_arrow_static.sh
index 97835f15db7..73aba450ed7 100755
--- a/r/inst/build_arrow_static.sh
+++ b/r/inst/build_arrow_static.sh
@@ -58,7 +58,7 @@ ${CMAKE} -DARROW_BOOST_USE_SHARED=OFF \
     -DARROW_COMPUTE=ON \
     -DARROW_CSV=ON \
     -DARROW_DATASET=ON \
-    -DARROW_DEPENDENCY_SOURCE=${ARROW_DEPENDENCY_SOURCE:-AUTO} \
+    -DARROW_DEPENDENCY_SOURCE=BUNDLED \
     -DARROW_FILESYSTEM=ON \
     -DARROW_JEMALLOC=${ARROW_JEMALLOC:-ON} \
     -DARROW_JSON=ON \
diff --git a/r/tools/linuxlibs.R b/r/tools/linuxlibs.R
index 760da6f92f4..1de5dc11998 100644
--- a/r/tools/linuxlibs.R
+++ b/r/tools/linuxlibs.R
@@ -288,8 +288,7 @@ build_libarrow <- function(src_dir, dst_dir) {
 
 ensure_cmake <- function() {
   cmake <- Sys.which("cmake")
-  # TODO: should check that cmake is of sufficient version
-  if (!nzchar(cmake)) {
+  if (!nzchar(cmake) || cmake_version() < 3.2) {
     # If not found, download it
     cat("**** cmake\n")
     CMAKE_VERSION <- Sys.getenv("CMAKE_VERSION", "3.16.2")
@@ -312,6 +311,18 @@ ensure_cmake <- function() {
   cmake
 }
 
+cmake_version <- function() {
+  tryCatch(
+    {
+      raw_version <- system("cmake --version", intern = TRUE, ignore.stderr = TRUE)
+      pat <- ".*?([0-9]+\\.[0-9]+\\.[0-9]+).*"
+      which_line <- grep(pat, raw_version)
+      package_version(sub(pat, "\\1", raw_version[which_line]))
+    },
+    error = function(e) return(0)
+  )
+}
+
 #####
 
 if (!file.exists(paste0(dst_dir, "/include/arrow/api.h"))) {