diff --git a/openmp/libomptarget/src/omptarget.cpp b/openmp/libomptarget/src/omptarget.cpp
--- a/openmp/libomptarget/src/omptarget.cpp
+++ b/openmp/libomptarget/src/omptarget.cpp
@@ -1127,11 +1127,13 @@
   /// Add a private argument
   int addArg(void *HstPtr, int64_t ArgSize, int64_t ArgOffset,
              bool IsFirstPrivate, void *&TgtPtr, int TgtArgsIndex,
-             const map_var_info_t HstPtrName = nullptr) {
+             const map_var_info_t HstPtrName = nullptr,
+             const bool AllocImmediately = false) {
     // If the argument is not first-private, or its size is greater than a
     // predefined threshold, we will allocate memory and issue the transfer
     // immediately.
-    if (ArgSize > FirstPrivateArgSizeThreshold || !IsFirstPrivate) {
+    if (ArgSize > FirstPrivateArgSizeThreshold || !IsFirstPrivate ||
+        AllocImmediately) {
       TgtPtr = Device.allocData(ArgSize, HstPtr);
       if (!TgtPtr) {
         DP("Data allocation for %sprivate array " DPxMOD " failed.\n",
@@ -1148,6 +1150,7 @@
 #endif
       // If first-private, copy data from host
       if (IsFirstPrivate) {
+        DP("Submitting firstprivate data to the device.\n");
         int Ret = Device.submitData(TgtPtr, HstPtr, ArgSize, AsyncInfo);
         if (Ret != OFFLOAD_SUCCESS) {
           DP("Copying data to device failed, failed.\n");
@@ -1325,13 +1328,16 @@
       TgtBaseOffset = 0;
     } else if (ArgTypes[I] & OMP_TGT_MAPTYPE_PRIVATE) {
       TgtBaseOffset = (intptr_t)HstPtrBase - (intptr_t)HstPtrBegin;
-      // Can be marked for optimization if the next argument(s) do(es) not
-      // depend on this one.
-      const bool IsFirstPrivate =
-          (I >= ArgNum - 1 || !(ArgTypes[I + 1] & OMP_TGT_MAPTYPE_MEMBER_OF));
+      const bool IsFirstPrivate = (ArgTypes[I] & OMP_TGT_MAPTYPE_TO);
+      // If there is a next argument and it depends on the current one, we need
+      // to allocate the private memory immediately. If this is not the case,
+      // then the argument can be marked for optimization and packed with the
+      // other privates.
+      const bool AllocImmediately =
+          (I < ArgNum - 1 && (ArgTypes[I + 1] & OMP_TGT_MAPTYPE_MEMBER_OF));
       Ret = PrivateArgumentManager.addArg(
           HstPtrBegin, ArgSizes[I], TgtBaseOffset, IsFirstPrivate, TgtPtrBegin,
-          TgtArgs.size(), HstPtrName);
+          TgtArgs.size(), HstPtrName, AllocImmediately);
       if (Ret != OFFLOAD_SUCCESS) {
         REPORT("Failed to process %sprivate argument " DPxMOD "\n",
                (IsFirstPrivate ? "first-" : ""), DPxPTR(HstPtrBegin));
diff --git a/openmp/libomptarget/test/mapping/lambda_by_value.cpp b/openmp/libomptarget/test/mapping/lambda_by_value.cpp
new file mode 100644
--- /dev/null
+++ b/openmp/libomptarget/test/mapping/lambda_by_value.cpp
@@ -0,0 +1,34 @@
+// RUN: %libomptarget-compilexx-run-and-check-generic
+
+#include <stdio.h>
+#include <stdint.h>
+
+// CHECK: before: [[V1:111]] [[V2:222]] [[PX:0x[^ ]+]] [[PY:0x[^ ]+]]
+// CHECK: lambda: [[V1]] [[V2]] [[PX_TGT:0x[^ ]+]] 0x{{.*}}
+// CHECK: tgt   : [[V2]] [[PX_TGT]] 1
+// CHECK: out   : [[V2]] [[V2]] [[PX]] [[PY]]
+
+int main() {
+  int x[10];
+  long y[8];
+  x[1] = 111;
+  y[1] = 222;
+
+  auto lambda = [&x, y]() {
+    printf("lambda: %d %ld %p %p\n", x[1], y[1], &x[0], &y[0]);
+    x[1] = y[1];
+  };
+
+  printf("before: %d %ld %p %p\n", x[1], y[1], &x[0], &y[0]);
+
+  intptr_t xp = (intptr_t) &x[0];
+#pragma omp target firstprivate(xp)
+  {
+    lambda();
+    printf("tgt   : %d %p %d\n", x[1], &x[0], (&x[0] != (int*) xp));
+  }
+  printf("out   : %d %ld %p %p\n", x[1], y[1], &x[0], &y[0]);
+
+  return 0;
+}
+