* Spain's 10-year bond yield tumbles
* Yield gap over Germany at tightest in a week at 118 bps
* Catalonia pressured to step back from independence pledge
* Spain's IBEX stock index outperforms peers
* Euro zone periphery govt bond yields http://tmsnrt.rs/2ii2Bqr (Updates throughout)
LONDON, Oct 9 (Reuters) - Spanish borrowing costs fell to a one-week low on Monday, narrowing the gap over top-rated Germany on hopes that Catalonia would this week take a step back from a unilateral declaration of independence from Spain.
Bigger-than-expected demonstrations in Barcelona on Sunday called for Catalonia to remain part of Spain, raising pressure on the regional government to back away from a declaration it could make as soon as Tuesday.
Moves by local companies to re-locate their headquarters elsewhere are also putting pressure on the region's pro-independence leader Carles Puigdemont to back down.
Puigdemont will address the Catalan parliament at 6 p.m. (1600 GMT) on Tuesday on "the current political situation" amid speculation he could ask the assembly to declare independence.
The Catalan region's head of foreign affairs, Raul Romeva, told the BBC on Friday that the Catalan parliament intended to make a decision on independence, without specifying when.
Madrid on Friday meanwhile apologised for the first time for police use of violence in trying to hinder a weekend referendum it had declared illegal, soothing financial market jitters.
"Some of the uncertainty has been reduced -- the tone from Puigdemont has become more conciliatory and (Spain's Prime Minister Mariano) Rajoy has also stepped back," said Peter Chatwell, head of euro rates strategy at Mizuho.
"It is expected that Puigdemont will on Tuesday say an independence bid in the future will be considered, which is a dramatic step back from expectations last week that a deceleration of independence is imminent."
Spain's 10-year bond yield tumbled 8 basis points to a 1-week low of 1.64 percent, down 17 bps from more than six-month highs hit last week.
That narrowed the gap over Germany's 10-year bond yield to a one-week low around 118 basis points -- having stretched out to 136 bps last week at the height of worries about a conflict between Catalonia and the central government in Madrid.
Spanish stocks rallied 0.7 percent on Monday, out performing broader European share markets, while the euro was a tad firmer.
There was also some relief after DBRS ratings agency late on Friday confirmed Spain's rating at A (low), following concerns that the tensions in Catalonia could hurt Spain's ratings outlook.
"DBRS's central scenario is that Spain remains united. This reflects legal and institutional safeguards that make a unilateral secession highly unlikely," the agency said in a note.
"However, uncertainty over Catalonia will continue in the near to medium term. A period of prolonged and elevated tension in Catalonia or political uncertainty in Spain at large could weigh negatively on the economy and public finances."
For now, the fall in Spanish bond yields dragged Portuguese and Italian yields about 4 bps lower each. Most other euro zone bond yields were slightly lower on the day.
For Reuters Live Markets blog on European and UK stock markets see reuters://realtime/verb=Open/url=http://emea1.apps.cp.extranet.thomsonreuters.bi z / c m s / ? p a g e I d = l i v e m a r k e t s
(Reporting by Dhara Ranasinghe; Editing by Matthew Mpoke Bigg)