Sept 20 (Reuters) - Freshworks Inc on Monday raised its target price range for a U.S. initial public offering, which could bring up the valuation of the business and customer engagement software company to nearly $9.6 billion. The San Mateo, California-based company expects to raise $969 million at the top end of its new price range of $32 to $34 per share, up from $28 to $32 per share earlier. Reuters reported in April that Freshworks, which competes with Salesforce.com Inc (CRM.N), could be valued at around $10 billion in a stock market debut. Freshworks" technology is used by more than 50,000 companies in 120 countries, including high-profile names such as Delivery Hero SE (DHER.DE), Vice Media and Swedish payments firm Klarna. Morgan Stanley, J.P. Morgan and Bofa Securities are the lead underwriters for the Freshworks offering. Freshworks, which was launched from the South Indian city of Chennai, will list on the Nasdaq under the symbol "FRSH", a regulatory filing showed.