diff --git a/src/htx.c b/src/htx.c
index bc2ecbdf3..a7623b6d9 100644
--- a/src/htx.c
+++ b/src/htx.c
@@ -502,6 +502,19 @@ struct htx_ret htx_xfer_blks(struct htx *dst, struct htx *src, uint32_t count,
 		if (type == HTX_BLK_UNUSED)
 			goto next;
 
+		/* Be sure to have enough space to xfer all headers in one
+		 * time. If not while <dst> is empty, we report a parsing error
+		 * on <src>.
+		 */
+		if (mark >= HTX_BLK_EOH && (type == HTX_BLK_REQ_SL || type == HTX_BLK_RES_SL)) {
+			struct htx_sl *sl = htx_get_blk_ptr(src, blk);
+
+			if (sl->hdrs_bytes != -1 && sl->hdrs_bytes > count) {
+				if (htx_is_empty(dst))
+					src->flags |= HTX_FL_PARSING_ERROR;
+				break;
+			}
+		}
 
 		sz = htx_get_blksz(blk);
 		info = blk->info;
diff --git a/src/mux_h2.c b/src/mux_h2.c
index ddaa75180..55c7619c5 100644
--- a/src/mux_h2.c
+++ b/src/mux_h2.c
@@ -5406,8 +5406,11 @@ static size_t h2_rcv_buf(struct conn_stream *cs, struct buffer *buf, size_t coun
 
 		htx_xfer_blks(buf_htx, h2s_htx, count, HTX_BLK_EOM);
 
-		if (h2s_htx->flags & HTX_FL_PARSING_ERROR)
+		if (h2s_htx->flags & HTX_FL_PARSING_ERROR) {
 			buf_htx->flags |= HTX_FL_PARSING_ERROR;
+			if (htx_is_empty(buf_htx))
+				cs->flags |= CS_FL_EOI;
+		}
 
 		buf_htx->extra = (h2s_htx->extra ? (h2s_htx->data + h2s_htx->extra) : 0);
 		htx_to_buf(buf_htx, buf);