diff --git a/llama-util.h b/llama-util.h
index 88ec28dca..e775f64cc 100644
--- a/llama-util.h
+++ b/llama-util.h
@@ -172,7 +172,7 @@ struct llama_mmap {
 #ifdef _POSIX_MAPPED_FILES
     static constexpr bool SUPPORTED = true;
 
-    llama_mmap(struct llama_file * file, bool prefetch = true) {
+    llama_mmap(struct llama_file * file, size_t prefetch = (size_t) -1 /* -1 = max value */) {
         size = file->size;
         int fd = fileno(file->fp);
         int flags = MAP_SHARED;
@@ -184,9 +184,9 @@ struct llama_mmap {
             throw std::runtime_error(format("mmap failed: %s", strerror(errno)));
         }
 
-        if (prefetch) {
+        if (prefetch > 0) {
             // Advise the kernel to preload the mapped memory
-            if (madvise(addr, file->size, MADV_WILLNEED)) {
+            if (madvise(addr, std::min(file->size, prefetch), MADV_WILLNEED)) {
                 fprintf(stderr, "warning: madvise(.., MADV_WILLNEED) failed: %s\n",
                         strerror(errno));
             }
diff --git a/llama.cpp b/llama.cpp
index 8312ffedf..6909db1ac 100644
--- a/llama.cpp
+++ b/llama.cpp
@@ -679,12 +679,16 @@ struct llama_model_loader {
 
     void load_all_data(llama_progress_callback progress_callback, void *  progress_callback_user_data, llama_mlock * lmlock) {
         size_t data_size = 0;
+        size_t prefetch_size = 0;
         for (const llama_load_tensor & lt : tensors_map.tensors) {
             data_size += lt.size;
+            if (lt.ggml_tensor->backend == GGML_BACKEND_CPU) {
+                prefetch_size += lt.size;
+            }
         }
 
         if (use_mmap) {
-            mapping.reset(new llama_mmap(&file_loaders.at(0)->file, false));
+            mapping.reset(new llama_mmap(&file_loaders.at(0)->file, prefetch_size));
             if (!lmlock) {
                 // Don't call the callback since the actual loading will be lazy
                 // and we can't measure it.
@@ -2317,7 +2321,7 @@ int llama_apply_lora_from_file_internal(struct llama_context * ctx, const char *
 
         // maybe this should in llama_model_loader
         if (model_loader->use_mmap) {
-            model_loader->mapping.reset(new llama_mmap(&model_loader->file_loaders.at(0)->file, /* prefetch */ false));
+            model_loader->mapping.reset(new llama_mmap(&model_loader->file_loaders.at(0)->file, /* prefetch */ 0));
         }
     }